tag:blogger.com,1999:blog-81550367665325936802024-03-04T20:47:16.727-08:00Cantor on the ShoreA Pop Set Theory BlogVincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.comBlogger25125tag:blogger.com,1999:blog-8155036766532593680.post-31223985592973738892016-05-08T23:00:00.001-07:002016-05-08T23:00:28.603-07:00Encyclopedia Wand and the End of Enthropy<i>Actually, this post was meant to capitalize on the fleeting moment of fame my blog had, being linked in the description of the first video below. Of course, my pace is totally different from the rest of the web, so while the "moment" lasted one week, it took me one month to write this post, so long for timeliness. Well, it's a nice post anyway. Enjoy!</i><br />
<br />
Hey, there. How're you doing? If you just stumbled here, statistically it means that you arrived from <a href="https://youtu.be/SrU9YDoXE88">Vsauce</a> video. Did you? If not, you should see it, it's pretty awesome. I'll wait. Yeah, yeah, don't worry, I'll wait, honest.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/SrU9YDoXE88/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/SrU9YDoXE88?feature=player_embedded" width="320"></iframe></div>
<br />
Done? Great.<br />
<br />
Anyway, if you are like me, now you'll have the following questions hovering inside your mind:<br />
<ul>
<li>There sure are many large cardinals! Where can I know more about them? (In this blog, of course)</li>
<li>Cantor on the Shore? That's a <i>silly</i> name. (Look at yours, <i>Chad. </i>Anyway, <a href="http://cantorontheshore.blogspot.it/2014/07/this-blog-will-blow-your-mind-aka.html">here is the explanation</a>, it's an homage to Haruki Murakami)</li>
<li>Oh yeah? Then why you've never posted anything about Murakami?</li>
</ul>
<div>
And here we are! <a href="https://en.wikipedia.org/wiki/Hard-Boiled_Wonderland_and_the_End_of_the_World">Hard-Boiled Wonderland and the End of the World</a> (Sekai no owari to Hādo-Boirudo Wandārando) is a 1985 novel by Haruki Murakami. It is about... you know unicorns, right? In the brain... and then there is the Town, like... the cyphering of minds, with minds... and yet the shadow... Listen, there are libraries, music and a bit of sex, so it's a Murakami book. Read it, it's fun!</div>
<div>
<br /></div>
<div>
Anyway, at a certain point the "encyclopedia wand" is introduced:</div>
<div>
<br /></div>
<div>
<span style="font-family: "courier new" , "courier" , monospace;">"The encyclopedia wand's a theoretical puzzle, like Zeno's paradox. The idea is t'engrave the entire encyclopedia onto a single toothpick. Know how you do it?"</span></div>
<div>
<span style="font-family: "courier new" , "courier" , monospace;">"You tell me."</span></div>
<div>
<span style="font-family: "courier new" , "courier" , monospace;">"You take your information, your encyclopedia text, and you transpose it into numerics. You assign everything a two-digit number, periods and commas included. 00 is a blank, A is 01, B is 02 and so on. Then after you've lined them all up, you put a decimal point before the whole lot. So now you've got a very long sub-decimal fraction, 0.173000631... Next, you engrave a mark at exactly that point along the toothpick, then 0.3333's got t'be a third of the way from the tip. You follow?"</span></div>
<div>
<span style="font-family: "courier new" , "courier" , monospace;">"Sure."</span></div>
<div>
<br /></div>
<div>
Zeno's paradox! You have <a href="http://cantorontheshore.blogspot.com/2014/07/the-fault-in-fault-in-our-stars-part-1.html">my attention</a>, Mr. Murakami. But this has nothing to do with Zeno's paradox for now. In fact, everything is finite. The encyclopedia text is finite, in the end you have a number with many many digit, but it's finite. And please please please don't be mad at me but <span style="font-size: x-small;">0.3333 is not one third,</span> <span style="font-size: x-small;">0.3333..., with infinite decimal digits, is.</span></div>
<div>
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgr4ZMDiopvYuVnU3Y9AJWThMtU5FmiBpSfT0EH2sAQsdrIHNJ9CTwIMnmzPdK_gOK-k7ufmtAtUHE-c89ULV1IPYfdc1wquKstKvMeo1ZeDJxn-d2gAIduiYK11VnuYw0LQQylJdMFfA/s1600/raw.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="224" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgr4ZMDiopvYuVnU3Y9AJWThMtU5FmiBpSfT0EH2sAQsdrIHNJ9CTwIMnmzPdK_gOK-k7ufmtAtUHE-c89ULV1IPYfdc1wquKstKvMeo1ZeDJxn-d2gAIduiYK11VnuYw0LQQylJdMFfA/s320/raw.gif" width="320" /></a></div>
<div>
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrDuCnQp8VuY8GYXYqvWOOXE9WWLnVuhuxVfy2ZBJmfUiO7J2MEdrBVGh2Fg5FoV_Ox3ub2xW0npXYMd1qiYmlCBXtXV1E7CZarqUJ_LhMiec7QqXxJ-zLQa9PjyzESZklEaveQzE-2Q/s1600/boing.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrDuCnQp8VuY8GYXYqvWOOXE9WWLnVuhuxVfy2ZBJmfUiO7J2MEdrBVGh2Fg5FoV_Ox3ub2xW0npXYMd1qiYmlCBXtXV1E7CZarqUJ_LhMiec7QqXxJ-zLQa9PjyzESZklEaveQzE-2Q/s320/boing.jpeg" width="320" /></a></div>
<div>
<br /></div>
<div>
Well, but that's really a cool thought experiment. You pick a toothpick, and with only <i>one</i> sign you have written an entire encyclopedia. Let's continue our reading:</div>
<div>
<br /></div>
<div>
<span style="font-family: "courier new" , "courier" , monospace;">"That's how you can fit data of any length in a single point on a toothpick. Only theoretically, of course. No existin' technology can actually engrave so fine a point. [...] Make the fraction as long as you want. It'll be finite, but <i>pretty</i> near eternal. Though if you make it a repeatin' decimal, why, then it <i>is</i> eternal."</span></div>
<div>
<br /></div>
<div>
So Murakami agrees with me that that was finite. The infinite part, now, is... disappointing. That's all? First of all, why limiting yourself to the repeatin' decimal? (I'd like to know the original, here, though, to see if there is something lost in translation.) A <a href="https://en.wikipedia.org/wiki/Repeating_decimal">repeatin' decimal</a> is a number whose digits (wait for it...) repeat themselves, over and over, to infinity. But there are all sorts of numbers with non repeatin' decimals, like <a href="https://en.wikipedia.org/wiki/Pi">pi</a> or <a href="https://en.wikipedia.org/wiki/E_(mathematical_constant)">e</a>, A repeatin' decimal (why am I writing like this?) actually does not have much information, it can be coded in a finite way (for example you can engrave in black when the decimal is finite, in red when it is repeatin', so 0.3333... is just 0.3 in red). To go <i>really</i> eternal you need to go irrational.<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOOUZjZb-RwEUuxjMaztyboTFuu4enUwoJeAN4UcBnU1MuGuo2XWMaVe44OrStpIhiMkjI546lbKxDHvNScKvOUpVgnQqB5zQffBwUN2nqeawj_K-0meGV-X3W01_77ncL2DxfPwQwYA/s1600/download.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOOUZjZb-RwEUuxjMaztyboTFuu4enUwoJeAN4UcBnU1MuGuo2XWMaVe44OrStpIhiMkjI546lbKxDHvNScKvOUpVgnQqB5zQffBwUN2nqeawj_K-0meGV-X3W01_77ncL2DxfPwQwYA/s320/download.jpg" width="215" /></a></div>
<br />
NO! I don't mean that! (One day I'll regret using all the image space of the blog for dumb jokes). Let me recap quickly, for those that were a bit sleepy in math class (we all went through that):<br />
<ul>
<li>Rational numbers are those that can be written as fractions, like 22/7, 1/3, 5/4 and so on. Some of this have a finite number of decimals, others infinite, but repeating (like 1/3=0.3333...)</li>
<li>Irrational numbers are all the other ones. </li>
</ul>
</div>
<div>
Irrational numbers, as they cannot be written in fractions, have all sorts of fun properties. For example, pick a circle:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGssiDjW3yZJchporssIjXQXZ_JiU-txhyWliJAhTv22vGpxgvRCiY0i726vGDvfhA7HAynqfIwy7AolHcWcDWj-Ol314JFvrByT-Kdhq0yzoPxdKVbXQiiyTeDVvsspA5aLp4cEhFtw/s1600/2000px-Circle_-_black_simple.svg.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGssiDjW3yZJchporssIjXQXZ_JiU-txhyWliJAhTv22vGpxgvRCiY0i726vGDvfhA7HAynqfIwy7AolHcWcDWj-Ol314JFvrByT-Kdhq0yzoPxdKVbXQiiyTeDVvsspA5aLp4cEhFtw/s200/2000px-Circle_-_black_simple.svg.png" width="200" /></a></div>
Suppose the circumference is 1m, Now pick your favorite irrational number, say square root of 2 (it works with any). Start from a point, and walk square root metres on the circle. You'll stop around here:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRMII25xgWLVCjBtxjhVgMHmK8RziDFCqsgJ-IT1QEG2nV4DVX5-935w_2lVUChol07vLdXo-3QvyK7AiHmySfm_T2oX90VhjL9I6me3-fa4pbQ6gjYfq626BO2dQM1USOYtZMnmZqAw/s1600/2000px-Circle_-_black_simple.svg+2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRMII25xgWLVCjBtxjhVgMHmK8RziDFCqsgJ-IT1QEG2nV4DVX5-935w_2lVUChol07vLdXo-3QvyK7AiHmySfm_T2oX90VhjL9I6me3-fa4pbQ6gjYfq626BO2dQM1USOYtZMnmZqAw/s200/2000px-Circle_-_black_simple.svg+2.png" width="200" /></a></div>
<br />
Now do it again, and again, and again, infinite times!<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVD3PpV9jOG9zevpCQnCNqZiWpogfVv-8EDVUrToxYVI68ceqDeVAPespI_J5CvE0KurB3_9qxINXJsIgtgGj0jKQBgd2DDeiL9vXjak1rdsWiHjU-qzC02RNCV6-AfzXORLym31rQBQ/s1600/Irrational+circle+ani.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVD3PpV9jOG9zevpCQnCNqZiWpogfVv-8EDVUrToxYVI68ceqDeVAPespI_J5CvE0KurB3_9qxINXJsIgtgGj0jKQBgd2DDeiL9vXjak1rdsWiHjU-qzC02RNCV6-AfzXORLym31rQBQ/s200/Irrational+circle+ani.gif" width="200" /></a></div>
All your stops are going to be uniformally distributed, i.e., they are going to cover pretty much all the circle in a uniform way, without areas that are more dense or less dense. This is called the <a href="https://en.wikipedia.org/wiki/Equidistribution_theorem">equidistribution theorem</a>. It's a special case of the ergodic theorem, the key theorem for <a href="https://en.wikipedia.org/wiki/Ergodic_theory">ergodic theory</a>. What is ergodic theory, you say? A picture is worth many words:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYwIZZyZiCXP3tuBdKPwxQ-NWdIrxQDN4m8xiADDCWV4p1Ndya4BPBq5DpUSS9-A5-JYKDL1YewRJ_U3VD1XujonTrx6duZutRB7I5Y0-A3XcDDEv77o2NB1xl0Y8aP9AXVIgSlN7ikA/s1600/Cyborg-Manual.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="208" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYwIZZyZiCXP3tuBdKPwxQ-NWdIrxQDN4m8xiADDCWV4p1Ndya4BPBq5DpUSS9-A5-JYKDL1YewRJ_U3VD1XujonTrx6duZutRB7I5Y0-A3XcDDEv77o2NB1xl0Y8aP9AXVIgSlN7ikA/s320/Cyborg-Manual.jpg" width="320" /></a></div>
<div style="text-align: center;">
<span style="font-size: x-small;">OK, sorry, I have no idea too. This is one of the results</span></div>
<div style="text-align: center;">
<span style="font-size: x-small;">in a Google Image search</span></div>
<div style="text-align: center;">
<span style="font-size: x-small;"><br /></span></div>
Let's go back to our encyclopedia wand now. We wrote an entire encyclopedia with one carving. Let's do now the opposite: we take the toothpick, we make one carving, and we ask: <i>what did we write?</i> Well, if we are doing it with an irrational number, prepare to be surprised! The same ergodic theorem above says that we have, with probability 1 (so not full certainty, but pretty much), everything. <i>Everything</i>.<br />
<br />
The full encyclopedia? It's there. Maybe not beginning at the first digit, or the 10th, or the 345891st, but it's there somewhere. If you are in a hurry, there is even the short version. The Bible? Check. The Lord of the Rings? It's there, with the Hobbit, the Silmarillion and all. The last Dan Brown? Also there, with all its 44 translations. Even with translations that were never published, like in <a href="https://en.wikipedia.org/wiki/Tinigua">Tinigua</a> and <a href="https://en.wikipedia.org/wiki/Malak-Malak_language">Mullukmulluk</a>.<br />
<br />
You know your favorite slash fiction between Severus Snape and the Teletubbies? Yep, still there. But wait! Then... also that fantastic book you had in mind for so long and never bothered to write it's... there? Even if it was never written? And also the story of your life, in all the details? The story of your future life, already written there, in that single irrational number, like you cannot even change it? I know what you're thinking, because there I can find the same exact thoughts you are having right now while you are reading this...<br />
<br />
Whoa whoa, there, let's stop the trip here. Better not to go into that rabbit hole. But yet, so much power in one tiny little number!<br />
<br />
There is something that it's bothering me, though, in Murakami's book. This phrase:<br />
<br />
<span style="font-family: "courier new" , "courier" , monospace;">Only theoretically, of course. No existin' technology can actually engrave so fine a point.</span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><br /></span>
Why so cautious, Murakami? Why chickenin' out like that? You wrote about <a href="https://en.wikipedia.org/wiki/After_the_quake">super-tall frogs</a> that want to destroy Tokyo while sipping tea, cat killer <a href="https://en.wikipedia.org/wiki/Kafka_on_the_Shore">whiskey icons</a>, parallel words populated by unicorns, <a href="https://en.wikipedia.org/wiki/Dance_Dance_Dance_(novel)">men that speak in tattoo</a>, and this is where you draw the line? This is what you consider too irrealistic, too bizarre, too-much-out-there? Engraving a fine point?<br />
<br />
If that's the case, then, pardon me. I'm going to make a flute out of cats' souls.<br />
<br />
<i>Thanks to Prof. Fuchino for having pointed this to me.</i><br />
<br />
<br /></div>
Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com2tag:blogger.com,1999:blog-8155036766532593680.post-17507009694226691872016-03-31T23:59:00.000-07:002016-04-07T13:14:53.621-07:00Stop everything you're doing and watch this infinity<br />
<span style="font-size: medium;">i.e., what if Buzzfeed took hold of this blog? </span><br />
<b><span style="font-size: medium;"><br /></span></b>
<b><span style="font-size: large;">1. Like, right?</span></b><br />
<ol>
</ol>
<iframe allowfullscreen="" class="giphy-embed" frameborder="0" height="360" src="//giphy.com/embed/l2R0a55Yzy5Erf128" width="360"></iframe><br />
<br />
<b><span style="font-size: large;">2. I have all the feels</span></b><br />
<br />
<iframe allowfullscreen="" class="giphy-embed" frameborder="0" height="360" src="//giphy.com/embed/l41m4IRJqzgtFQ4O4" width="360"></iframe><br />
<br />
<b><span style="font-size: large;">3. I can't even</span></b><br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgH94qAOP2BiPw-775lHzoTpG9aMFkEW5J2Jz7V4eC-2ffBYd5YL6UjxzZ94lsuELJjnByAgvedlEksffTL_L5Yi4xAgADagh7jrv5xhdWTspAYPK1JTAnsVhvNv6eRNGw2s4PZYSEZZg/s1600/650px-Omega-exp-omega-labeled.svg.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgH94qAOP2BiPw-775lHzoTpG9aMFkEW5J2Jz7V4eC-2ffBYd5YL6UjxzZ94lsuELJjnByAgvedlEksffTL_L5Yi4xAgADagh7jrv5xhdWTspAYPK1JTAnsVhvNv6eRNGw2s4PZYSEZZg/s320/650px-Omega-exp-omega-labeled.svg.png" width="260" /></a></div>
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<span style="font-size: x-small;"><br /></span>
<span style="font-size: x-small;">By Pop-up casket (talk); original by User:Fool - Own work, CC0, <a href="https://commons.wikimedia.org/w/index.php?curid=22276515">https://commons.wikimedia.org/w/index.php?curid=22276515</a></span><br />
<br />
<b><span style="font-size: large;">4. This one, though</span></b><br />
<br />
<iframe allowfullscreen="" class="giphy-embed" frameborder="0" height="192" src="//giphy.com/embed/XkDyB1jCZMxUs" width="360"></iframe><br />
<br />
<b><span style="font-size: large;">5. This one will make you thirsty</span></b><br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisteQ-cpa67pDa57KIXymymP-uoxnk64ESsoNcZogBqKX1MNRmZtTiRxXfW2-JR0vDe2p_lYdvwNkBK58MQFzuxiMA121fKiAh-u9ksZtfXvtAH5sSZLYtV3FovQbB6StEQjNCb14FjA/s1600/Diagonal_argument_01_svg.svg.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisteQ-cpa67pDa57KIXymymP-uoxnk64ESsoNcZogBqKX1MNRmZtTiRxXfW2-JR0vDe2p_lYdvwNkBK58MQFzuxiMA121fKiAh-u9ksZtfXvtAH5sSZLYtV3FovQbB6StEQjNCb14FjA/s1600/Diagonal_argument_01_svg.svg.png" /></a></div>
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<span style="font-size: x-small;"><br /></span>
<span style="font-size: x-small;">By Jochen Burghardt - Own work, CC BY-SA 3.0, <a href="https://commons.wikimedia.org/w/index.php?curid=30402203">https://commons.wikimedia.org/w/index.php?curid=30402203</a></span><br />
<span style="font-size: x-small;"><br /></span>
<b><span style="font-size: large;">6. OMG what</span></b><br />
<br />
<iframe allowfullscreen="" class="giphy-embed" frameborder="0" height="240" src="//giphy.com/embed/yoJC2KOr9xOgxQVSZa" width="360"></iframe><br />
<br />
<b><span style="font-size: large;">7. smh</span></b><br />
<br />
<iframe allowfullscreen="" class="giphy-embed" frameborder="0" height="160" src="//giphy.com/embed/VoqeTRR4KxNWo" width="360"></iframe><br />
<br />
<b><span style="font-size: large;">8. mind=blown</span></b><br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzUml4N0PcjSfHo5FY7CwSKWb9jVTJMuxGn9bVx4GxMtj15dxge4g53Itp3y_ShABs6nyQQFRHBGvtH0S7YEZgfL0zVQVen1QNKdVkdRTSrZoESjKSuBgOyow5V5GCZoPz7FwCGUA-4w/s1600/bsp.gif" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="160" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzUml4N0PcjSfHo5FY7CwSKWb9jVTJMuxGn9bVx4GxMtj15dxge4g53Itp3y_ShABs6nyQQFRHBGvtH0S7YEZgfL0zVQVen1QNKdVkdRTSrZoESjKSuBgOyow5V5GCZoPz7FwCGUA-4w/s320/bsp.gif" width="320" /></a></div>
<br />
<br />Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com1tag:blogger.com,1999:blog-8155036766532593680.post-20850423235013837032015-11-02T13:17:00.002-08:002015-11-02T13:17:40.895-08:00What is this Boole's Google Doodle?I imagine the scene. You wake up this morning (yes, you), dizzy after a brave night of questionable choices, you fire up Google, and here it is! A new doodle! What is it?<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEin0UJ9fhUugnbX2Jqk2L1pIwPM-DL5Bozd1LntS72f2og5whuOuFt2qKzTiTTPie1pD4pHSpVJCNnEQm0Xc69FDVGci3_uGyfTgSHy4Txb9TADIbyjSPOVVoVRQKJp5GFrd5t84ERcbw/s1600/Booledoodle.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="97" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEin0UJ9fhUugnbX2Jqk2L1pIwPM-DL5Bozd1LntS72f2og5whuOuFt2qKzTiTTPie1pD4pHSpVJCNnEQm0Xc69FDVGci3_uGyfTgSHy4Txb9TADIbyjSPOVVoVRQKJp5GFrd5t84ERcbw/s320/Booledoodle.png" width="320" /></a></div>
<br />
Wait! There is the play button! Maybe it's another game! Like that time Pac-Man's Doodle made the world losing <a href="http://www.pcworld.com/article/197130/googles_pacman_cost_120_million_in_productivity.html">120 million dollars</a> in productivity! (Well, maybe this is <a href="http://www.businessinsider.com/no-the-google-pac-man-doodle-didnt-cost-anyone-anything-2010-5?IR=T">not true</a>). Or like the Halloween one, where you collected candy avoiding bats and ghosts. You passed the day collecting yellow candies. (Because you are a despicable person. Blue should have won!). Let's play!<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjtE-60p9G7CRxJZyKoVR0fzuV60iOMt4be0m3ZPxy23PnDIrcBmzuR6YMW1yxVwYE_XZwV4CAb_Hkh-7jWOCQVCtaIQFC1Fi0tF4HZaDNuTQ6c1-VElfgno7qrMIiX45jOFNXE9xIEw/s1600/BVddNj.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjtE-60p9G7CRxJZyKoVR0fzuV60iOMt4be0m3ZPxy23PnDIrcBmzuR6YMW1yxVwYE_XZwV4CAb_Hkh-7jWOCQVCtaIQFC1Fi0tF4HZaDNuTQ6c1-VElfgno7qrMIiX45jOFNXE9xIEw/s320/BVddNj.gif" width="320" /></a></div>
<br />
Huh. What's that? Blinking... stuff? How do you play? You cannot. Green, red, yellow... Maybe it's a traffic light? You try to do the robot in rhythm, but no, it doesn't make sense. You click on it, and it goes to the search results page of "George Boole". Humph! "Who is this guy?" you say, reddened from rage "That's... that's... that's <i>booloney</i>, Google!" (The questionable choices are really catching up on you).<br />
<br />
Yes, probably this doodle is not the most perspicuous. The people who knows who Boole is, I guess won't be too much excited by it, and <b>the people who don't know the adjective <i>boolean</i> will not understand what's going on</b>. Well, if you stumbled on this page with this question in mind, I'm here to help you!<br />
<br />
So, <b>the doodle of today is to celebrate the 200 years of <a href="https://en.wikipedia.org/wiki/George_Boole">George Boole</a>'s birth</b>. Who is George Boole? This guy:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjix80z_fQ4t_uvgBmZSAh8FDR0uz37tSU0ofAuGbi3ElljQkIjnZAF2XyyR6ziRW7R_NxTPJxSdVKYuMJKWMSOLslg6XttVcvyNfRm0STaP9OFEHzocEcdDl0TT3nZ0hrP623_fRrfsQ/s1600/George_Boole_color.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjix80z_fQ4t_uvgBmZSAh8FDR0uz37tSU0ofAuGbi3ElljQkIjnZAF2XyyR6ziRW7R_NxTPJxSdVKYuMJKWMSOLslg6XttVcvyNfRm0STaP9OFEHzocEcdDl0TT3nZ0hrP623_fRrfsQ/s320/George_Boole_color.jpg" width="238" /></a></div>
<br />
Boole, when it was not busy cultivating his magnificent sideburns (i.e., rarely), was a mathematician. A rather classical one, actually, he collected some results on differential equations and analysis that are still used. Yet, nothing to google-doodle about. His most important works are instead a small pamphlet called Mathematical Analysis of Logic, and the big budget sequel <a href="https://en.wikipedia.org/wiki/The_Laws_of_Thought">An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities</a> (hopefully the book is longer than the title). His idea was to pick Aristotelian logic and systematizing it, <b>formalizing</b> it. Conjunctions and disjunctions (and, or...) were not considered anymore grammatical structures, but <i>mathematical</i> ones. They were operations between sets. It was huge: Boole just <b>kicked out the philosophers from logic, and made it a mathematical concept</b>, in fact an algebraic one, therefore making possible all sorts of applications (and <b>computer science</b> is just one of the most famous).<br />
<br />
<b>How does it work</b>? It's called <a href="https://en.wikipedia.org/wiki/Boolean_algebra">Boolean algebra</a>. In the most basic instance, it's just "true" and "false", and the operations AND, OR and NOT between them.<br />
- AND is like multiplication: true AND true is true, all the others are false, e.g. false AND true is false<br />
- OR is more or less like addiction: false OR false is false, all the others are true, e.g. true OR false is true<br />
- NOT is clear: NOT true is false, and NOT false is true.<br />
<br />
Now see again the doodle: it's just <b>showing this operations</b>! When x appears, it means x is true, otherwise x is false. So the "G" (i.e., x AND y) lights up only when both x and y appear, the "l" (NOT x) only when x does not appear, and so on. There is another operation I haven't talked about: XOR. It's the "exclusive or", and x XOR y only if x or y appear, but not at the same time.<br />
<br />
Wait, there's more! If you substitute "true" and "false" with "on" and "off", then you have how all the circuits in all the computers function. They are true physical manifestations of boolean operations: if you open your laptop* you look into it, you will find XOR gates, AND gates, and so on.<br />
<br />
*I'm not going to be responsible for this.<br />
<br />
But is this logic? <b>Why are they called like conjuctions and disjunctions</b>? In a more complicated instance, AND, OR and NOT are operations among sentences. If you have two sentences, "sentence A" and "sentence B", then "sentence A AND sentence B" is the conjunction of the two. That is... "sentence A and sentence B". Whoah, deep.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6MNjtuPCAQR9ErcPKEThW6iEe-yESommcci-BnFoJszVMRU71X1QfXsFKiyXQeS_XWXIOYoeRsjAuTeWd-TfkDDzYAI-BJPrQELJSMflTpBo6_PxlH4lpZ_UVvzYJoxTXfu7XqEzxqw/s1600/confused1.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="213" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6MNjtuPCAQR9ErcPKEThW6iEe-yESommcci-BnFoJszVMRU71X1QfXsFKiyXQeS_XWXIOYoeRsjAuTeWd-TfkDDzYAI-BJPrQELJSMflTpBo6_PxlH4lpZ_UVvzYJoxTXfu7XqEzxqw/s320/confused1.jpeg" width="320" /></a></div>
<br />
I don't think that clarified anything. The parallel is: "sentence A AND sentence B" is true if and only if both sentences are true, and x AND y is true if and only if x is true and y is true. So the operations respect the logic, and the logic defines the operations: AND is really just "and", OR is just "or", but instead of connecting sentences, they give you the truth value of the sentences connected. Move on.<br />
<br />
You can see <a href="http://cantorontheshore.blogspot.co.at/2014/08/dont-be-naive-google.html">here</a> that <b>boolean operations are also operations on sets</b>. AND is just the intersection, OR the union, XOR the symmetric difference. Try it on Google! Search for "Google is awesome" AND "Yog-Sothoth evocation ritual", and then for "Google is awesome" OR "Yog-Sothoth evocation ritual". Now! Or else! (<a href="http://cantorontheshore.blogspot.co.at/2014/08/infinity-times-infinity-updates.html">Google is watching you trying</a>). So you know now why a search engine is so grateful to Boole (and why you should always have in handy the mystic scimitar of Barzai).<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0ZkAHnZQIPbVHysBg2qDmtvEJQRYUzR7cBuf3qM1IitrJ0Uz4wlUw-BK75TaPuiLXJxycgYrCdUQ3JLQwCxGWaze4Eh7pq-517543bU0D8QMU18Df2aIoONFdxb5nYJmQhfHdPqyUrw/s1600/cthulhu4prez-preview1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0ZkAHnZQIPbVHysBg2qDmtvEJQRYUzR7cBuf3qM1IitrJ0Uz4wlUw-BK75TaPuiLXJxycgYrCdUQ3JLQwCxGWaze4Eh7pq-517543bU0D8QMU18Df2aIoONFdxb5nYJmQhfHdPqyUrw/s320/cthulhu4prez-preview1.png" width="320" /></a></div>
<div style="text-align: center;">
<span style="font-size: x-small;">Don't be less evil</span></div>
<br />
There are many "boolean" things around. The aforementioned boolean algebras, but also boolean circuits, <b>boolean expressions</b> (those things in Excel that permits you to do magic), boolean functions, boolean models, boolean processors... Even <b>a crater on the Moon named Boole</b>!<br />
<br />
Boolean algebras, by the way, are essential in Set Theory. They are the basic of the forcing method, the paradigm-changing method that in recent years permitted to prove many independency results, therefore establishing once and for all that mathematics is incomplete, many questions have no answer. It's easy: build a boolean algebra that does the trick, then find an ultrafilter of it (it should be generic, mind you!), then in the generic extension...<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-y537LCmQ3TtgPFMEtqa2cIVnZs-hxcCBOI6t4kLhI5L80NZpguM1eO1PI94Iw9WVre1yQOQhfmS_aa4MaGeKA7OcPn2ikcK1dx6oDr0JxzerhBRv1JAtwLUwTtweh0GvvDeBHh7IFg/s1600/confused1.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="213" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-y537LCmQ3TtgPFMEtqa2cIVnZs-hxcCBOI6t4kLhI5L80NZpguM1eO1PI94Iw9WVre1yQOQhfmS_aa4MaGeKA7OcPn2ikcK1dx6oDr0JxzerhBRv1JAtwLUwTtweh0GvvDeBHh7IFg/s320/confused1.jpeg" width="320" /></a></div>
<br />
<br />
You know what? This is not the time to talk about this. This needs a whole other posts. So that's it, for now! Enjoy your Boole day AND have fun! OR NOT! Ha!<br />
<br />
(wait for it...)<br />
<br />
XOR.<br />
<br />
(I think I blew it)Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com0tag:blogger.com,1999:blog-8155036766532593680.post-45110300633140470442015-06-22T02:31:00.003-07:002015-06-22T02:31:36.360-07:00The Truth about The Zero Theorem<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjiOk_qLa54xsjVqNnzfzLg-ssdr6hmmCOD84BS9oqtgWmEuZEWO6422QQUHZ-a53iEUQKUxmxiaP8cu5kEWXirLEfI9wMTBnBahI2BOwgYoqm3B4xyOoT4_-v_GtKKnZbLF8IQ8jfT3w/s1600/435733-the-zero-theorem-the-zero-theorem-poster-art.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjiOk_qLa54xsjVqNnzfzLg-ssdr6hmmCOD84BS9oqtgWmEuZEWO6422QQUHZ-a53iEUQKUxmxiaP8cu5kEWXirLEfI9wMTBnBahI2BOwgYoqm3B4xyOoT4_-v_GtKKnZbLF8IQ8jfT3w/s320/435733-the-zero-theorem-the-zero-theorem-poster-art.jpg" width="216" /></a></div>
<br />
<br />
The other day I have seen The Zero Theorem, a recent movie by Terry Gilliam. The movie was not really appreciated by the critics, but if you are reading here, probably you have already seen it, so it doesn't really matter. Also, it was greatly imaginative, and Gilliam's fans will appreciate it. But if you've seen the movie, then you could ask: how much of it is mathematically true? Of course, it's all pretty surreal and it must not be taken at face value, but is there a kernel of truth? Maybe. Let's see.<br />
<b><br /></b>
<b>Q. What is the Zero Theorem? Is it something mathematicians are really trying to prove?</b><br />
A. The movie is rather vague, so I cannot know for certain. I have three hypotheses:<br />
<br />
The <a href="https://en.wikipedia.org/wiki/Rational_root_theorem">rational zero theorem</a>, or rational root theorem, is a classical algebra theorem that indicates all the possible rational solutions to a polynomial equation. Now there is a rule: if there is a proof of it on Wikipedia, it means that in the spectrum of sureness, it is between death and the protests following a new Facebook layout:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVoHjyzxM3HsPSDKIHuavcrb_GaSO_wxUgVybHGesqf06d8WusFKm-GxSFGpdv7UaryFBNGguorcX5UhcXNm-YA1v31dUzZ3rPBJk1ouSzaqveEgG2NE4xKnIErTNg_hoyL6X-u4DLnw/s1600/Certainty+Scale.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="266" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVoHjyzxM3HsPSDKIHuavcrb_GaSO_wxUgVybHGesqf06d8WusFKm-GxSFGpdv7UaryFBNGguorcX5UhcXNm-YA1v31dUzZ3rPBJk1ouSzaqveEgG2NE4xKnIErTNg_hoyL6X-u4DLnw/s400/Certainty+Scale.png" width="400" /></a></div>
<br />
so probably this is not what Gilliam had in mind.<br />
<br />
In the movie, Bob tries to explain what is going on: proving the Zero Theorem would be proving the <a href="https://en.wikipedia.org/wiki/Big_Crunch">Big Crunch</a>, that is the cosmological idea that the universe at a certain point will stop expanding and will start to collapse, ultimately disappearing. This is opposed to the <a href="https://en.wikipedia.org/wiki/Future_of_an_expanding_universe">Big Freeze</a>, where the universe will continue to expand and therefore cool off, at a certain point too cold for life, and the <a href="https://en.wikipedia.org/wiki/Big_Rip">Big Rip</a>, where the expanding universe will rip, and two points will be at infinite distance between each other. Whatever notorious B.I.G. the scientists are trying to prove, probably it cannot be solved just with a theorem, it needs practical experiments. So, yeah, probably that is what Gilliam had in mind, even if it doesn't hold much water. Yet there is space for another hypothesis...<br />
<br />
At a certain point we learn that Qohen managed to prove that 0 equals 93.78926%, and the objective is 100%.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5oIUrNcu3aGFOemyubp8GXoTwCZm5Yc03ifywBhpHMMgjEIIgKZYZ9ztxqPdXUeVkd4OL1Q2djd-206WAjyAw8Wt618CHDNn7Iev77lscAnneK5UfFLBc6DhAM6tAE_2EmDhzmn1a8g/s1600/0%253D100%2525.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="182" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5oIUrNcu3aGFOemyubp8GXoTwCZm5Yc03ifywBhpHMMgjEIIgKZYZ9ztxqPdXUeVkd4OL1Q2djd-206WAjyAw8Wt618CHDNn7Iev77lscAnneK5UfFLBc6DhAM6tAE_2EmDhzmn1a8g/s320/0%253D100%2525.jpg" width="320" /></a></div>
<div style="text-align: center;">
<span style="font-size: x-small;">The <a href="https://www.youtube.com/watch?v=Hcywf9mwF5U">Mainframe</a>! I must access the Mainframe!</span></div>
<br />
100% of what? What about... everything? Then yes, that would be bad, it would destroy all mathematics. There is a catch, though: it would not be a slow climbing like in the movie, one percentage point at the time, but a sudden death! One just needs a very small percentage of things to be 0, and then everything would be 0. <b>In fact, one needs just 1=0</b>. Think about it: if 1=0, then 2=1+1=0, and 3=0, and so on, every number is 0. Also, if you have a triangle, what would be its area? 0, so it's a point, so any geometric shape would not exists. Using the <a href="https://xkcd.com/704/">principle of explosion</a> (yes, I linked a xkcd comic and not the Wikipedia page, good enough) one could falsify any theorem. In fact, it's an inside joke between set theorists that 1=0 is the strongest axiom, because it can prove everything and its opposite (and this also answers the final question of one <a href="http://cantorontheshore.blogspot.co.at/2014/10/one-reinhardt-and-counting.html">previous post</a> of mine! this is an eight-months <a href="https://en.wikipedia.org/wiki/Chekhov%27s_gun">Chekov's gun</a>!)<br />
<br />
<b>Q. Is it possible to prove that everything is 0? Can it really happen?</b><br />
A. Yes, it can definitely happen. If you read this <a href="http://cantorontheshore.blogspot.co.at/2014/09/artemis-fowl-and-large-cardinals_22.html">post</a>, then you'd know that mathematics is a list of things we say are true (axioms) and everything that can be logically derived by them. Gödel proved with his <a href="https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems">Second Incompleteness Theorem</a> that we cannot be sure that such a list doesn't prove a contradiction, ever. <b>Tomorrow one smart guy can wake up, and prove that all mathematics is 0</b>.<br />
<br />
<b>Q. What would happen if the Zero Theorem is true? Everything would be meaningless, like in the movie?</b><br />
<b><br /></b>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWnygK_-xDet7VGCLpVN1cxdtoqznX3U_Cvq-Q0Or-DrI2E7RjgTuMREaiOTUfYpFkrnw3TBNLFeoetV2-JjL2YXrBnV1YHPvj94OnWpmDz-8DvcCLrJzuSayleSebpIIqpqHOT2sm0g/s1600/3768044_orig.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="173" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWnygK_-xDet7VGCLpVN1cxdtoqznX3U_Cvq-Q0Or-DrI2E7RjgTuMREaiOTUfYpFkrnw3TBNLFeoetV2-JjL2YXrBnV1YHPvj94OnWpmDz-8DvcCLrJzuSayleSebpIIqpqHOT2sm0g/s320/3768044_orig.jpg" width="320" /></a></div>
<b><br /></b>
A. Well, my life surely. Your life will probably be fine. Sure, a lot of the things we do depend on mathematics, but there is maybe a way to save it, <b>it depends how deep the contradiction is</b>. What brings a contradiction? Is it very complex formulas? Then maybe limiting the complexity of formulas we are safe (again, we can <i>never</i> be sure). Too big numbers? Then we can limit the numbers (here you are again, ultrafinitist! You happy, now?). A contradiction is like gangrene (yikes, what a bad metaphor): sure, it can infect all the body, but you can cut just the part affected. If it is the little toe, then mathematics will be pretty much unscathed. Of course, it can happen that the problem is in the heart, for example in the basic arithmetic of numbers. That would mean that the basic mathematics does not work like we predict, for example <b>computers could not work as expected and we would need a team of mathematicians that put order in the disorder</b> (like in the movie) and rethink mathematics. But it is very implausible,<br />
<br />
<b>Q. Is the name of the protagonist relevant to mathematics?</b><br />
A. There is a Cohen that is extremely important in mathematics, <a href="https://en.wikipedia.org/wiki/Paul_Cohen_(mathematician)">Paul Cohen</a>, his work on the continuum problem (see <a href="http://cantorontheshore.blogspot.co.at/2015/01/infinite-mess-part-two.html">this post</a>) was unprecedented and changed mathematics completely, thanks to his technique, called forcing. But this has nothing to do with the Zero Theorem, and as the surname of Qohen is Leth, probably Gilliam was referring to Qoheleth, the <a href="https://en.wikipedia.org/wiki/Ecclesiastes">Ecclesiastes</a>.<br />
<br />
<b>Q. What is the Transfinite Paradox?</b><br />
A. Yes, at the beginning Qohen solves the Transfinite Paradox. Transfinite is how mathematicians call an infinite that is not absolute, but all such paradoxes are already solved, so I have no idea what he meant.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbEUQxLC5LoI8MBG5CY1C028hyphenhyphen6CJ4jw7U02yZv5_PXlOtyPr6J3QvEVitjiSrmOUVI4XctPmNRCOwELng1ASaEMd796Fo3huMQRnVylcZ0x6b6FPouVgNXPzUPkeXmgMkAu3u5Fiomg/s1600/maxresdefault+%25281%2529.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbEUQxLC5LoI8MBG5CY1C028hyphenhyphen6CJ4jw7U02yZv5_PXlOtyPr6J3QvEVitjiSrmOUVI4XctPmNRCOwELng1ASaEMd796Fo3huMQRnVylcZ0x6b6FPouVgNXPzUPkeXmgMkAu3u5Fiomg/s320/maxresdefault+%25281%2529.jpg" width="320" /></a></div>
<br />
Also, can we please, please, <i>please </i>avoid every time there is someone doing something mathematical avoid to picture him (because it's always a male) as a disturbed, asocial and in general a few cards short of a deck? Of course, mathematicians are a odd bunch, just like musicians, artists, writers and every job where you need <b>a lot of passion to survive</b>. But there are many ways to be odd, and not only this Asperger-like grumpiness: probably the worst case, bordering libel, was in the recent <i>The Imitation Game</i>, where they used the usual palette to paint Turing, that in reality was a fun, sociable and likeable person. Come on, moviemakers! De-Sheldonize mathematics!<br />
<br />
<i>Infinity is a staple of science fiction, like in <a href="http://cantorontheshore.blogspot.co.at/2014/07/rendezvous-with-futurama.html">Futurama</a> or <a href="http://cantorontheshore.blogspot.co.at/2014/09/artemis-fowl-and-large-cardinals.html">Artemis Fowl</a>. But if you want to know more about its limits, then go no further then <a href="http://cantorontheshore.blogspot.co.at/2014/10/one-reinhardt-and-counting.html">here</a>.</i>Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com0tag:blogger.com,1999:blog-8155036766532593680.post-85503109959129442642015-05-18T01:00:00.000-07:002015-05-18T09:08:15.588-07:00An Answer to the Ultrafinitist Below<i>This post is an answer to <a href="http://cantorontheshore.blogspot.co.at/2015/04/some-infinities-dont-exist-more-than.html">this one</a> (yes, I am answering myself, please wait before calling the mental house). If you read that one, you probably noticed that the style was, um, different. That is because that should have been an April fools' joke (you could guess it because the words fools and joke were bold, and because it was the 1st of April). It was a collection of arguments that people against infinity use. I wanted to answer immediately, but <a href="https://www.youtube.com/watch?v=SkWeMvrNiOM">life finds a way</a>... to kill your plans. So let's answer them now.</i><br />
<i><br /></i>
Who is right, then? I wrote 20 posts about infinity, but the arguments against it are pretty convincing, aren't they? Well, in fact thery are not really <i>wrong</i>, they are just too partial. Let's see them.<br />
<br />
<b>Q. Is it true that the universe is not infinite?</b><br />
A. I am not a physicist, so I cannot answer this question in all the details (that's a bad start). <a href="http://www.esa.int/Our_Activities/Space_Science/People/Is_the_Universe_finite_or_infinite_An_interview_with_Joseph_Silk">In this interview</a> Prof. Joseph Silk says that simply we don't know. It can be either way, with our current knowledge it's not possible to know. According to <a href="http://en.wikipedia.org/wiki/Universe#Shape_of_the_Universe">Wikipedia</a>, one of the models of the Universe with most consensus is the <a href="http://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric">Friedmann–Lemaître–Robertson–Walker metric</a>, that appears to be infinite, but I have no idea. Anyway, it seems like it doesn't matter, The time we have here is finite, therefore even if the Universe is infinite, we will be able to observe just a finite part of it.<br />
<br />
<b>Q. But is it true that between two points there is just a finite amount of space?</b><br />
A. Again, we don't know. I found this nice discussion on <a href="http://physics.stackexchange.com/questions/33273/is-spacetime-discrete-or-continuous">StackExchange</a>, and the models we are using now to calculate stuff do have infinite points between two points (are continuous). The consensus seems to be that this is the case, but we cannot know.<br />
<br />
<b>Q. Ok, so we don't know if the infinite exists in this Universe, and maybe we will never know. But is it used in physics practice?</b><br />
A. Constantly. The language of physics is mathematics, and there infinity is a necessary tool. If you think about it, nothing in mathematics really <i>exists</i>. There is no real number, there are no points, no lines. Triangles do not exist.<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLzdEiDDIKzYsuN31PUzoKEVUEL3XzOvUM8dErfz0DrLE-dE_e2hPj3ezxlsiwEdtLCs5CUYGHjVyFi8ygkYxlUjDT38Nfa075_lZXj_c5VBWOl-pyw2fh5uLnYkWGnFFcyNZtDGiqGw/s1600/Sad+triangle.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="237" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLzdEiDDIKzYsuN31PUzoKEVUEL3XzOvUM8dErfz0DrLE-dE_e2hPj3ezxlsiwEdtLCs5CUYGHjVyFi8ygkYxlUjDT38Nfa075_lZXj_c5VBWOl-pyw2fh5uLnYkWGnFFcyNZtDGiqGw/s320/Sad+triangle.png" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: left;">
And yet we use them every day, because they work.</div>
<br />
<b>Q. Therefore infinity is used in maths.</b><br />
A. That is not a question. Anyway, yes. math is practically <i>founded</i> on the infinite. Leibniz, Newton & co., while founding <a href="http://en.wikipedia.org/wiki/Calculus#Limits_and_infinitesimals">calculus</a>, were using it just like any other quantity. Now we know better, and we actually <i>can</i> get rid of it, decide a maximum number and stay there. But the problem is: which one?<br />
<br />
As an example, let's take computer science. In 1936 Turing invented the <a href="http://en.wikipedia.org/wiki/Turing_machine">Turing machine</a>, that is not a real machine, but a hypothetical one that is infinite and that can mimic the behaviour of <i>any</i> computer. It has been a fundamental tool for the understanding of computer science (well, it was the <i>invention</i> of the idea of computer). Now suppose that Turing, instead of conceiving an infinite machine, invented a <i>finite</i> one. How big do you think it would have been? 2000 possible numbers? 20000?<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgY7r0JX0kJKYoR6TjVrWHNVK6oAlJRvddLaYrYWykidSUHD1LDA5kH4CGHyOpnsQGmcd9JWW41PegPmmb09MbJrFMSwLciAiL4jy-69PB0WANQ2pwCdKpFLZs86vNv4GDDxCGBAe9XKg/s1600/Z3_Deutsches_Museum.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgY7r0JX0kJKYoR6TjVrWHNVK6oAlJRvddLaYrYWykidSUHD1LDA5kH4CGHyOpnsQGmcd9JWW41PegPmmb09MbJrFMSwLciAiL4jy-69PB0WANQ2pwCdKpFLZs86vNv4GDDxCGBAe9XKg/s320/Z3_Deutsches_Museum.jpg" width="320" /></a></div>
<div style="text-align: center;">
<span style="font-size: x-small;">"<a href="http://commons.wikimedia.org/wiki/File:Z3_Deutsches_Museum.JPG#/media/File:Z3_Deutsches_Museum.JPG">Z3 Deutsches Museum</a>" by <a class="extiw" href="https://de.wikipedia.org/wiki/User:Venusianer" title="de:User:Venusianer">Venusianer</a> at the <a class="extiw" href="https://de.wikipedia.org/wiki/" title="de:">German language Wikipedia</a>. Licensed under <a href="http://creativecommons.org/licenses/by-sa/3.0/" title="Creative Commons Attribution-Share Alike 3.0">CC BY-SA 3.0</a> via <a href="https://commons.wikimedia.org/wiki/">Wikimedia Commons</a>.
</span></div>
<div style="text-align: center;">
<span style="font-size: x-small;">The Z3, built in 1941, could memorize only 64 words of 22 bit</span></div>
<br />
It was surely unconceivable at that time more space, but now we can easily and cheapily have 137438953472 numbers hidden under our nails. If the Turing machine was finite, every year a bunch of people would have to meet to raise its size. Instead, we have a reliable and universal way to represent all computers, past and future, thanks of infinity. For example, now we now if something is just <a href="http://en.wikipedia.org/wiki/Halting_problem">not possible to calculate with a computer</a>. If we just deal with finite stuff, one can say "Heh, maybe if our computers are bigger/faster, we can calculate that", and spend millions of real world money for an impossible task.<br />
<br />
The same is true for mathematics. Deciding a limit number is ridiculous. Also many concepts first start with infinity, and then become finite. Without the infinity part, we probably just wouldn't have such concepts, as "calculus" or "computer".<br />
<br />
<b>Q. Can we imagine the infinite?</b><br />
A. That's an interesting question!<br />
<b>Q. Thank you.</b><br />
A. I mean, even <a href="http://cantorontheshore.blogspot.co.at/2014/08/advanced-thinking-thinking-infinity.html">kids</a> know perfectly well what infinity is, right? Yet, we cannot possibly think all the numbers. Our brain is finite. But we can think of the <i>totality</i> of the numbers. For example, think of a glass of water. In that glass, say, there are 8 x 10^24 molecules of water (that is more than 8 septillions for an American). Our mind cannot possibly think of each molecule, there are too many. Yet, you had no problem in thinking of a glass of water, am I correct?<br />
<br />
So, that's it! Infinity is safe and alive! You can continue to infinity-and-beyond your stuff, people! Oh, one more thing...<br />
<br />
<b>Q. Are all finitists so obnoxious, like in the previous post?</b><br />
A. Yes. Well, that's unfair, just the more vocal ones. I don't know what's wrong with them. At the mere hint of infinity, they unleash an anger and spite usually reserved to breast-feeding forums and Ben Affleck's Batman. They call people who use infinity "mystics", "religious" or even "fetishists". They stalk all famous mathematicians, ready to attack (I am not famous, but I do have a finitist that comes into my office). They feel like the whole world is wrong, and they're frustrated that no one listens to them, before it's too late. They are not stupid or ignorant, mind you, sometimes they are brilliant thinkers. It's just that we work on a different fundamental assumption, and there is no way to please both. But can we nonetheless aceept each other and be friends? Please?<br />
<br />
<br />
<br />
<br />
<br />
<br />Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com0tag:blogger.com,1999:blog-8155036766532593680.post-26150212212936900642015-04-01T00:32:00.000-07:002015-07-25T02:21:21.045-07:00Some infinities don't exist more than other infinities (an ultrafinitist point of view)<i>Attention! This was an April Fools' post, so it says the exact opposit of what I think. Still, it's food for thought, and I answer to this in <a href="http://cantorontheshore.blogspot.co.at/2015/05/an-answer-to-ultrafinitist-below.html">this post</a>.</i><br />
<br />
Hahaha. Infinity.<br />
<br />
What a<b> joke</b>.<br />
<br />
Only<b> fools</b> would really believe that such a thing exists. Don't you see it? It's just a ruse to make you question the reality you are seeing with your own eyes! Wake up sheeple!<br />
<br />
Infinity is just an illusion. <a href="http://en.wikipedia.org/wiki/God_is_dead">God is dead</a>, so why we don't do the same to infinity, huh? Time to get ride of it. Well, we are already doing it. We thought the universe was infinite, well, <a href="http://www.newscientist.com/article/dn4250-tantalising-evidence-hints-universe-is-finite.html#.VRqr2_msWUI">not so much</a>. We thought the speed of light was infinite, <a href="http://en.wikipedia.org/wiki/Speed_of_light">not even close</a>. We thought that between two points there were infinite points, or that between two instants there were infinite moments of time, wrong <a href="http://en.wikipedia.org/wiki/Planck_length">again</a> and <a href="http://en.wikipedia.org/wiki/Planck_time">again</a>. Everything around us is finite. You don't go to the pet shop and buy infinite kittens. That would be too cuteness for anybody.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNsRSUvMzfgUQ72MRdKpbfy-Yj55cSy0rH04N_yaJqByfzxKYDRos_s1a-gliisu1s6q97g9wYnklSYjlswFs6pv06TyzM5P_vL9ZxmHQcCqaj6jpVsTKFdyfEdbM7DNvvm7RbHqi6og/s1600/lolbrary.com_39692_1386352032.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="247" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNsRSUvMzfgUQ72MRdKpbfy-Yj55cSy0rH04N_yaJqByfzxKYDRos_s1a-gliisu1s6q97g9wYnklSYjlswFs6pv06TyzM5P_vL9ZxmHQcCqaj6jpVsTKFdyfEdbM7DNvvm7RbHqi6og/s1600/lolbrary.com_39692_1386352032.gif" width="320" /></a></div>
<div style="text-align: center;">
<span style="font-size: x-small;">Spoiler alert: this is not possible.</span></div>
<div style="text-align: left;">
<span style="font-size: x-small;"><br /></span></div>
<div style="text-align: left;">
Then love is infinite? Sorry lovey-dovey couples, at the very least one of you will die, so you can just start to delete that tattoo right away. But then death is infinite? No, it's not: everybody is always dead for a finite amount of time. Julius Caesar has been dead for 2059 years and 17 days, Marvin Gaye for exactly 31 years. Give it up, infinitists!</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
So what about the numbers? Are they infinite? Of course not. Just pick the number of particles in the universe, its time and calculate all the possible combinations of them. It will be a finite number, the biggest on. What do you say? What about that number +1? Pfft, that does not <i>exist</i>. Just because you can <i>imagine</i> a number, it doesn't mean it exists. I can imagine a unicorn, but still not receiving it for Christmas. Because it does not exist.</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
Ok, so you're saying that one can imagine infinite numbers. Come on, try it. Start: 10, 100, 1000... At a certain point you cannot go on anymore. Some numbers are so huge, that we cannot conceive them, But then, where do they exist, if not in nature or in anybody's mind? Answer: they don't.</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
So you like tortoises and Achilleses, right? If you think about it, the solution is easy: the world is like a Monopoly table. There will be a moment when Achille and the Tortoise will be one step close. Then the tortoise should do half a step, but it cannot, so it stays, and Achille does one step and reaches the tortoise. Easy peasy!</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
Enough with this infinite foolishness, then! Eat your vegetables, and embrace the world as it is! (Vegetables are delicious, too! Especially the Brussels Sprouts.)</div>
<div style="text-align: left;">
<br /></div>
Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com2tag:blogger.com,1999:blog-8155036766532593680.post-9269973203051364262015-01-25T23:31:00.002-08:002015-01-25T23:31:19.449-08:00Infinite Mess (Part Two)<i>This is the second part of a post that started <a href="http://cantorontheshore.blogspot.co.at/2015/01/infinite-mess-part-one.html">here</a>. Go read it, if you still haven't done so. Or don't, and try to guess what is going on. It'll be enlightening anyway.</i><br />
<br />
So, what did DFW wrote(*)? Here is the source, verbatim:<br />
<br />
<span style="font-family: Courier New, Courier, monospace;">The Continuum Hypothesis gets characterized in all kind of different ways: [...</span><span style="font-family: 'Courier New', Courier, monospace;">]</span><span style="font-family: 'Courier New', Courier, monospace;">(**)</span><span style="font-family: 'Courier New', Courier, monospace;"> Is </span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhEVcMNQm2XLGSeT-bJ1gPIqnEGwj5YW95DVeU2wVf9dHGOHMkeAgJ4sFxPPsemodsjcBNhCkvUvf1rz_Ixu5QvM9bRMqbfRWWi3ope0XacCQEU-kflmz9VPKLHabCDjvv3kLxUUwHxnw/s1600/c.png" imageanchor="1" style="clear: left; display: inline !important; font-family: 'Courier New', Courier, monospace; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhEVcMNQm2XLGSeT-bJ1gPIqnEGwj5YW95DVeU2wVf9dHGOHMkeAgJ4sFxPPsemodsjcBNhCkvUvf1rz_Ixu5QvM9bRMqbfRWWi3ope0XacCQEU-kflmz9VPKLHabCDjvv3kLxUUwHxnw/s1600/c.png" /></a><span style="font-family: 'Courier New', Courier, monospace;">the same as </span><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiv455QxUoL4BykVqhC54xJCoGFNbySVNDcywU6y-agqpFZRMmdCpTDHJhh3AAlNFl628Ci5Of-f53QRsDK1REli-o_l0KnHp74hHq5NVtPI6m-BOqY8Rc4EPsUMCS2dV_jiqqYUVUbNg/s1600/2toaleph0.png" imageanchor="1" style="clear: left; display: inline !important; font-family: 'Courier New', Courier, monospace; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiv455QxUoL4BykVqhC54xJCoGFNbySVNDcywU6y-agqpFZRMmdCpTDHJhh3AAlNFl628Ci5Of-f53QRsDK1REli-o_l0KnHp74hHq5NVtPI6m-BOqY8Rc4EPsUMCS2dV_jiqqYUVUbNg/s1600/2toaleph0.png" /></a><span style="font-family: 'Courier New', Courier, monospace;">? </span><br />
<br />
Let me first try to explain to you what is the Continuum Hypothesis, let's see if I am better in this than Foster Wallace. First, let's get the objects straight:<br />
<br />
<b><a href="http://en.wikipedia.org/wiki/Natural_number">Natural numbers</a></b> are the numbers like 0, 1, 2 and so on. Basically, if you hear a number and you can imagine the same amount of zebras (or any other object, but I prefer zebras), it's a natural number. You can't imagine pi zebras, or 1.25 zebras. This are the numbers that we all learned in school, and I really shouldn't have spent 67 words on this.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbjWfF-siKJX1dAwqMlG8hwAreA625P1KajAHRclKmrLmdvdkX0DM2nD1DEZYvHABi8DUJra8XgTH0OB6B9Z_4fVGo7hTIQVQsiyQXO-GNN0wuH_Tp0xEIzBPfLTxZhF8Y3UBPAgoxFg/s1600/numbers.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbjWfF-siKJX1dAwqMlG8hwAreA625P1KajAHRclKmrLmdvdkX0DM2nD1DEZYvHABi8DUJra8XgTH0OB6B9Z_4fVGo7hTIQVQsiyQXO-GNN0wuH_Tp0xEIzBPfLTxZhF8Y3UBPAgoxFg/s1600/numbers.jpg" height="205" width="320" /></a></div>
<div style="text-align: center;">
<span style="font-size: x-small;">And one picture</span></div>
<br />
<b><a href="http://en.wikipedia.org/wiki/Real_number">Real numbers</a></b> are all the numbers. Period. Even with infinite digits after the digital point. 2, 3.45, square root of 2, pi are all real numbers.<br />
<br />
Now, Cantor proved that some infinities are bigger than other infinities, right? Well, he was more specific: he proved that the real numbers are more than the natural numbers. Pretty cool. Then he asked, in his <a href="http://www.digizeitschriften.de/dms/img/?PPN=PPN243919689_0084&DMDID=dmdlog15&LOGID=log15&PHYSID=phys248#navi" style="font-style: italic;">Ein Beitrag zur Mannigfaltigkeitslehere</a>, whether the real numbers are immediately larger than the natural numbers, or if there is something in between, larger than the natural number but smaller than the real numbers. This is called <b>Continuum Problem</b>, because continuum was fancy -talk for real numbers. Then Cantor said: you know what? I think that there is nothing in between. This is called <b>Continuum Hypothesis</b>, and as it is formulated, it's just an opinion, a hypothesis (hence the name): we don't know the answer to the continuum problem, so let's suppose it's this.<br />
<br />
So? Understood? If you read DFW's book I'd really like to know if it was easier to understand than this.<br />
<br />
You can already see the first problem: the Continuum Hypothesis asserts something, so it cannot be a question! DFW is confusing it for the Continuum Problem. Already annoying. But let's go on: what are those strange symbols in DFW's quote?<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhEVcMNQm2XLGSeT-bJ1gPIqnEGwj5YW95DVeU2wVf9dHGOHMkeAgJ4sFxPPsemodsjcBNhCkvUvf1rz_Ixu5QvM9bRMqbfRWWi3ope0XacCQEU-kflmz9VPKLHabCDjvv3kLxUUwHxnw/s1600/c.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhEVcMNQm2XLGSeT-bJ1gPIqnEGwj5YW95DVeU2wVf9dHGOHMkeAgJ4sFxPPsemodsjcBNhCkvUvf1rz_Ixu5QvM9bRMqbfRWWi3ope0XacCQEU-kflmz9VPKLHabCDjvv3kLxUUwHxnw/s1600/c.png" /></a></div>
is the size of the set of the real numbers, i.e., how many real numbers there are.<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiv455QxUoL4BykVqhC54xJCoGFNbySVNDcywU6y-agqpFZRMmdCpTDHJhh3AAlNFl628Ci5Of-f53QRsDK1REli-o_l0KnHp74hHq5NVtPI6m-BOqY8Rc4EPsUMCS2dV_jiqqYUVUbNg/s1600/2toaleph0.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em; text-align: center;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiv455QxUoL4BykVqhC54xJCoGFNbySVNDcywU6y-agqpFZRMmdCpTDHJhh3AAlNFl628Ci5Of-f53QRsDK1REli-o_l0KnHp74hHq5NVtPI6m-BOqY8Rc4EPsUMCS2dV_jiqqYUVUbNg/s1600/2toaleph0.png" /></a></div>
<br />
<span style="text-align: center;">is the size of the set of the sets of natural number (I'll stop you before you start </span><a href="http://knowyourmeme.com/memes/xzibit-yo-dawg" style="text-align: center;">Xzibit memes</a><span style="text-align: center;">), i.e., how many sets of natural numbers there are.</span><br />
<br />
DFW is showing something completely different than the Continuum Problem, then, he's asking if there is the same quantity of real numbers and sets of integers, like it is a great mistery.<br />
<br />
It is not! It's Set Theory 101: they are the same! It's not a mistery, it's almost trivial! What were you thinking, DFW? <b>He got everything completely confused, he wanted to show the Continuum Hypothesis and he showed an exercise for students that involves objects that have nothing to do with the Continuum Hypothesis.</b><br />
<br />
You want to know why they are the same? Mmm... this is not immediate, unfortunately one has really to write down a mathematical proof. So put on your favorite thinking hat...<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjsnygglGf4150i2yYXvVYExg-HNcDBuIFr1qejtQ0uKCukO1BVfRcWVUc-xkgE1BkFBd4ERjGsQaiFZeIeHNOidID4f6n5ncKgWSsCx7E61xfXO5PEcnZOtlrDb6zfm03zAXIgDD0swg/s1600/weird-hats-13.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjsnygglGf4150i2yYXvVYExg-HNcDBuIFr1qejtQ0uKCukO1BVfRcWVUc-xkgE1BkFBd4ERjGsQaiFZeIeHNOidID4f6n5ncKgWSsCx7E61xfXO5PEcnZOtlrDb6zfm03zAXIgDD0swg/s1600/weird-hats-13.jpg" height="320" width="320" /></a></div>
<div style="text-align: center;">
<br /></div>
<div style="text-align: center;">
<span style="font-size: x-small;">This is mine. Don't judge me, the situation at the office is awkward enough.</span></div>
<br />
and concentrate on the following (***).<br />
<br />
To prove that two sets have the same size, one should be able to connect every object of one set to only one other object on the other set,<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWM1cx8RJBZ5kh4UEYfYVs6FWwZ_7fKIfpZl3J1wDW67neSnGxReRa2CPI2Hhsdr4ZTaXSsz1tEYvpemtCi3ia_gJ3_TZ6AOw8Y9g0HTAKPeBnO4j5ZhFoH89mw0moJG-y0zBOw6Rd0A/s1600/Aftercomparison.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWM1cx8RJBZ5kh4UEYfYVs6FWwZ_7fKIfpZl3J1wDW67neSnGxReRa2CPI2Hhsdr4ZTaXSsz1tEYvpemtCi3ia_gJ3_TZ6AOw8Y9g0HTAKPeBnO4j5ZhFoH89mw0moJG-y0zBOw6Rd0A/s1600/Aftercomparison.jpg" height="213" width="320" /></a></div>
<div style="text-align: center;">
<span style="font-size: x-small;"><a href="http://cantorontheshore.blogspot.co.at/2014/07/the-fault-in-fault-in-our-stars-part-1.html">In this way</a></span></div>
<div style="text-align: center;">
<span style="font-size: x-small;"><br /></span></div>
<div style="text-align: left;">
Try as you might, this doesn't work in this case. What to do then? Mathematicians know a weird simple trick (doctors HATE it!!): what if we connect all the apples to the oranges and there are oranges left? This means that the oranges cannot be less than the apples, right? (****) And if we connect, in another way, all the oranges to the apples and there are apples left, this means that the apples cannot be less than the oranges! So they are the same quantity!</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
Let's see first that the real numbers cannot be less than the sets of natural numbers. Pick a set of natural numbers</div>
<div style="text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5IKXiSb1wYXr3JkMIRUWhr4gqoqZ3oZf3CSUUoH7ICfydCbeE-zn7uk2iQtsHds4LsGxtURNCEkVi4PRqBupLTgbhZ_PFrhflArsJ8RDx9Blm7nqi2G2u1tKP_3YVBv7tbCJg2v_gFA/s1600/setofnaturalnumbers.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh5IKXiSb1wYXr3JkMIRUWhr4gqoqZ3oZf3CSUUoH7ICfydCbeE-zn7uk2iQtsHds4LsGxtURNCEkVi4PRqBupLTgbhZ_PFrhflArsJ8RDx9Blm7nqi2G2u1tKP_3YVBv7tbCJg2v_gFA/s1600/setofnaturalnumbers.png" height="36" width="320" /></a></div>
<div style="text-align: left;">
then draw it in the number line</div>
<div style="text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhwTfxZmCAk5XCRo-V50YdFLTKsNFoxFbiopS_JSVr7nsIv4MJXIJ1cQFEE4ALZAyUjEwLzgUovoflnFvOYGXzb_MAkMFscy1FFT-AFpljPqDgWUhqtjXX5wwtIday_XzNcPVMC3d0qhA/s1600/setofnaturalnumbers3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhwTfxZmCAk5XCRo-V50YdFLTKsNFoxFbiopS_JSVr7nsIv4MJXIJ1cQFEE4ALZAyUjEwLzgUovoflnFvOYGXzb_MAkMFscy1FFT-AFpljPqDgWUhqtjXX5wwtIday_XzNcPVMC3d0qhA/s1600/setofnaturalnumbers3.png" height="58" width="320" /></a></div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
write a 0 when it's empty, 1 when it's full</div>
<div style="text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuuwirNe8w7KSgQVLbO1VnVrjHQQfO7oGQ06cK4mOSPuxEq4bpM0Pfv6UKpGofFp5-TIBmY8WyPqlWOBepZtlxlCa_HVt9urQjIfccuvAEoJBg-cFbdwsnmjwvIg7Ocqx8ZBHLk093LA/s1600/setofnaturalnumbers1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuuwirNe8w7KSgQVLbO1VnVrjHQQfO7oGQ06cK4mOSPuxEq4bpM0Pfv6UKpGofFp5-TIBmY8WyPqlWOBepZtlxlCa_HVt9urQjIfccuvAEoJBg-cFbdwsnmjwvIg7Ocqx8ZBHLk093LA/s1600/setofnaturalnumbers1.png" height="30" width="320" /></a></div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
and finally add 0. at the front.</div>
<div style="text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgAVrjsx9Li80EhW9avma8eQrmXx2djnKFpbV0Z_LgUZ1CiUwBQmaCiP-z_Tp4PuSKKFcCRZrFvt95gXlPKmtPAyz__7XdGi0z7u6nRUMOgZog6zkz-HfNkAEzunfN74zSOt2te9AW3Yw/s1600/setofnaturalnumbers2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgAVrjsx9Li80EhW9avma8eQrmXx2djnKFpbV0Z_LgUZ1CiUwBQmaCiP-z_Tp4PuSKKFcCRZrFvt95gXlPKmtPAyz__7XdGi0z7u6nRUMOgZog6zkz-HfNkAEzunfN74zSOt2te9AW3Yw/s1600/setofnaturalnumbers2.png" height="25" width="320" /></a></div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
There you are, for each set of natural number, you can write a different real number. So real numbers cannot be less.</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
Now to see that the set of natural numbers cannot be less than the real numbers, I won't describe it, I'll just show it:</div>
<div style="text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhatkoZAPZ0oxJdF1G742w-65DweYA0ikyV6VIJY40N7twSy93WiyS1zrft8PMP0uZINJ07CwfzWD75J7FuJ08AXZ9e4qT1LsIsZrVdD7UfTDdPs7Sp5Ys62FlGeyf0GAUylta4A0LYSA/s1600/fromrealtoset.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhatkoZAPZ0oxJdF1G742w-65DweYA0ikyV6VIJY40N7twSy93WiyS1zrft8PMP0uZINJ07CwfzWD75J7FuJ08AXZ9e4qT1LsIsZrVdD7UfTDdPs7Sp5Ys62FlGeyf0GAUylta4A0LYSA/s1600/fromrealtoset.png" height="130" width="400" /></a></div>
<div style="text-align: left;">
Therefore the two things do have the same size, and the climax of DFW's booklet is an epic fail.</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
Really, it's disappointing. It started so well, with Zeno's paradox of the turtle and everything...</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
Wait a minute, where did I see the turtle paradox and Cantor work together i<a href="http://cantorontheshore.blogspot.co.at/2014/07/the-fault-in-fault-in-our-stars-part-1.html">n the wrong way</a>? John Green is a <a href="http://johngreenbooks.com/copyright-and-david-foster-wallaces-commencement-address/">big fan of David Foster Wallace</a>, right? Maybe...</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
Now that I think about it, the book-in-the-book An Imperial Affliction has many things in common with Infinite Jest, like the non-ending. But oh! Of course! The writer of AIA, Van Houten, is so similar to the prose of DFW! His obscurity, his way of talking encyclopedic but hard to understand... and it is Van Houten that connects (wrongly) the turtle paradox and Cantor's Theorem! Also, John Green has surely read Everything and More, he even reviewed it for Booklist Magazine!</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
<b>John Green has put Cantor's Theorem in The Fault in Our Stars because he read it on Everything and More, and he did it wrongly because it was confusing already in the original book!</b> That's where everything starts! That's a scoop!</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
What? What are you saying? You mean... he already admitted that in the FAQ <a href="http://johngreenbooks.com/questions-about-the-fault-in-our-stars-spoilers/">page I have already linked</a> once? </div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
Oh.</div>
<div style="text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_ykufSPxSEfpoOMPiy1YdPFp81fsep_Xweem7FDcMxfAVVJ5JUYL92Oybhcu9isCbRtysQ0fUjVbqBVoTv2denHSf59LwFGTgv2X6zeo3FKvOhpaDjnjj5WHYHfU-KAGg8fI6o-GcrA/s1600/cute-sad-kitten06.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_ykufSPxSEfpoOMPiy1YdPFp81fsep_Xweem7FDcMxfAVVJ5JUYL92Oybhcu9isCbRtysQ0fUjVbqBVoTv2denHSf59LwFGTgv2X6zeo3FKvOhpaDjnjj5WHYHfU-KAGg8fI6o-GcrA/s1600/cute-sad-kitten06.jpg" height="288" width="320" /></a></div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
I'll see myself out.</div>
<br />
(*) Since I am taking for granted that you, reader, are a DFW fan, I am adding lots of footnotes. Have fun!<br />
(**) I skipped the other three characterizations of Foster Wallace, without context it's pretty useless. For the curious: one is wrong and the other two are characterizations of the Continuum Problem, not the Continuum Hypothesis.<br />
(***) Or don't. Really, you can just skip the whole paragraph, you, reader, are the king, because I am so post-modern.<br />
(****) In the finite case, it means that the apples are less than the oranges, but infinity is <i>weird</i>.<br />
<br />
<i>Thanks to Gabriel for pointing this out to me.</i>Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com2tag:blogger.com,1999:blog-8155036766532593680.post-31349038233790355552015-01-12T01:17:00.000-08:002015-02-24T09:29:51.178-08:00Infinite Mess (Part One)Uh-oh. That's it. I'm going to do it. I am going to criticize a very well-beloved author, an author that touched the hearts of millions of people, one of the most influential and innovative writers of the last 20 years, according to the Los Angeles Times, also one that met a tragic and untimely end, and therefore untouchable.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgiHvCMMXv9FedOpa84X7oDvF1b8__bWtIueMXOgUBDilbowyn8QFR2zC1KmDlXJbM82b7ImNJU4EvQJqZ68c6VC0bH7U3YImbIcvuc4HZGnfIqAThQIVQg12dRzlsyW_xhMzOY0hJWsw/s1600/universal-frankenstein-angry-mob.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgiHvCMMXv9FedOpa84X7oDvF1b8__bWtIueMXOgUBDilbowyn8QFR2zC1KmDlXJbM82b7ImNJU4EvQJqZ68c6VC0bH7U3YImbIcvuc4HZGnfIqAThQIVQg12dRzlsyW_xhMzOY0hJWsw/s1600/universal-frankenstein-angry-mob.jpg" height="212" width="320" /></a></div>
<br />
<div style="text-align: center;">
<span style="font-size: x-small;">The reactions of literary fans are known to be sober</span></div>
<br />
I'm not going to be only critical, I am going to <i>destroy</i> one of his works (well, at least some lines). I am talking about <a href="http://en.wikipedia.org/wiki/David_Foster_Wallace">David Foster Wallace</a>, the brilliant mind behind <a href="http://en.wikipedia.org/wiki/Infinite_Jest">Infinite Jest</a>, an encyclopedic, metamodernist, <a href="http://en.wikipedia.org/wiki/Hysterical_realism">hysterical realist</a> novel that almost single-handedly put him in the curriculum of English literature courses. He's edgy, irreverent, inventive and also sweet, how can I possibly go against him (especially since he cannot defend himself)? Not only that, but I'll even claim that I am better than him in explaining some stuff! Oh my, some little blogger really went over his head, now.<br />
<br />
I will use his words to defend myself:<br />
<br />
<span style="font-family: Courier New, Courier, monospace;">The Mentally Ill Mathematician seems now in some ways to be what the Knight Errant, Mortified Saint, Tortured Artist, and Mad Scientist have been for other eras: sort of our Prometheus, the one who goes to forbidden places and returns with gifts we can all use but he alone pays for. </span><br />
<br />
Well, here I am; as a Prometheus (and maybe Mentally Ill Mathematician, as this blog seems to attest) I can be forgiven if I bash a literary genius, as I am also bringing gifts for everybody, guys! They come from forbidden places!<br />
<br />
Unfortunately, it seems that Foster Wallace didn't go where I've been. In 2003 he wrote a booklet, <a href="http://en.wikipedia.org/wiki/Everything_and_More_(book)">Everything and More</a>, about the history of infinity, and especially the work of our good old Cantor. Great, right? Finally some popular recognition to our hero! So, is it any good?<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgacyG9V2YIejJbU04hxRDJJzq93fDk4-271Jgt80oHwfrnpFbC4dgdfBekxV927PjjMDCwG_A6wL6cbO-JAG2E4uI1dsEKzMrD5E0dJ1WXhUsX2RkwccDGRXx1PvAAgyRWz48hU4qroA/s1600/Everything_and_More_cover.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgacyG9V2YIejJbU04hxRDJJzq93fDk4-271Jgt80oHwfrnpFbC4dgdfBekxV927PjjMDCwG_A6wL6cbO-JAG2E4uI1dsEKzMrD5E0dJ1WXhUsX2RkwccDGRXx1PvAAgyRWz48hU4qroA/s1600/Everything_and_More_cover.jpg" height="320" width="210" /></a></div>
<br />
<br />
Disclaimer: I haven't read it. What I read are the critical reviews of <a href="http://www.cs.sjsu.edu/faculty/rucker/wallace_review.pdf">Rudy Rucker</a> and <a href="http://www.ams.org/notices/200406/rev-harris.pdf">Michael Harris</a>, the second one being really interesting as it is more forgiving to the author, and some other snippet caught here and there in some preview, It was enough to bum me. <b>There is always a misunderstanding when writers try to explain mathematical concepts</b>: the literary way of dealing with concepts is through <b>vagueness</b>. The beauty of a poem is that the words carry with them many, many meanings, and elicit in our mind different responses, therefore being able with this overlap to create sensations that would be impossible to explain in plain words. Mathematics is the opposite: its beauty is in the <b>perfection</b> of its concepts. It's a huge mechanism full of cogs and wheels, and yet <b>everything works perfectly</b>, every minimal part is on time. It's like juggling, or <a href="https://www.google.com/search?q=rock+balancing&rlz=1C1TEUA_enAT506AT506&espv=2&biw=1455&bih=705&tbm=isch&tbo=u&source=univ&sa=X&ei=k0GpVO3ULoW2UdCNhNAL&ved=0CB8QsAQ">rock balancing</a>: the slightest error can ruin everything.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDw98IQ5jCHGcK2Udzx14AbK0Od8Lum2yLlYiv9kzap-a40HIpvrUtlsTNqWBLaHwMHfwplUWTujOQ97xYZxGpSHmahhpX84HbFWYUMcwaUXpGHIsP32B_LWfOybC7ZP6-oIa2MnpLHw/s1600/LiteraturevsMathematics.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDw98IQ5jCHGcK2Udzx14AbK0Od8Lum2yLlYiv9kzap-a40HIpvrUtlsTNqWBLaHwMHfwplUWTujOQ97xYZxGpSHmahhpX84HbFWYUMcwaUXpGHIsP32B_LWfOybC7ZP6-oIa2MnpLHw/s1600/LiteraturevsMathematics.jpg" height="161" width="320" /></a></div>
<br />
<div style="text-align: center;">
<span style="font-size: x-small;">Literature vs Mathematics</span><br />
<span style="font-size: xx-small;">(Left: (c) <a href="http://www.photographersgallery.com/photo.asp?id=2854">Frank Grisdale</a>)</span></div>
<div style="text-align: center;">
<br /></div>
<div style="text-align: left;">
You can see, then, how treating mathematical concepts with literary tools just doesn't work. It's even worse in this case: DFW is aggressively post-modern, and one of the staples of aggressive post-modernism is the <b>unreliability</b> of the narrator. And he admits that in this book! So, <b>what's the point of reading a mathematical book where the writer always tries to trick you?</b> Not only that, but he's trying to invent a new style, called "pop technical writing", that uses tons of footnotes (of course) and abbreviations that no one ever used. A complete mess.<br />
<br />
Well, then, that is <i>how</i> it is written. But what about the content? Surely there is a lot to learn reading this book, right?<br />
<br />
No.<br />
<br />
<b>Unfortunately, there are many, many errors.</b> There are <a href="http://www.thehowlingfantods.com/dfw/images/enmerrata.pdf">websites</a> that list all of them. There is enough material to publish one post per day, for a year. Now I cannot do that, I have already very few readers without actively alienate them with DFW marathons, so I'll choose just one. But it's a big one. Imagine: you spent hours and hours trudging through this book, through all the ridiculous fake notations, through hundreds of pages on convergent series, when finally you are realizing that the book <i>had</i> a direction, that all of this was to reach a particular point, You turn the page ready to read it, no, to <i>experience</i> it...<br />
<br />
and it is wrong.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/-w-58hQ9dLk?feature=player_embedded' frameborder='0'></iframe></div>
<div style="text-align: center;">
<span style="font-size: x-small;">The musical equivalent</span></div>
<div style="text-align: left;">
<span style="font-size: x-small;"><br /></span></div>
</div>
What? You don't expect me to explain the climax now, do you? Just go <a href="http://cantorontheshore.blogspot.com/2015/01/infinite-mess-part-two.html">here</a> to read about this crime against infinity.Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com0tag:blogger.com,1999:blog-8155036766532593680.post-55654700565678385272014-12-30T02:07:00.001-08:002015-02-24T09:27:19.951-08:00And the best tattoo of the year 2014 is...Ok, I am a standard mathematician. I don't want to reinforce the trite stereotype of the nerd math-lover, but I have thick glasses, often the colours of my clothes don't match (and my clothes are too big/too small/perfect just for old people), I learned how to bike waaay after learning the difference between mass and weight, and of course I am terrified, terrified by needlespleasepleasepleasetakeoutfrommyviewthatinstrumentofterror. I'm hopeless, you really wouldn't expect me to dissert about the current fashion in tattoos, would you?<br />
<br />
Yet, <a href="http://www.buzzfeed.com/alessiasantoro/2014s-most-popular-ink#.wupgWX618">Buzzfeed</a> made a list of the most trendy tattoos of the year 2014. And what do we have here?<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisyq4ZBhFlV4alTv_H3BJ-eYfXPtVgzSMHvhKa_3wD63gfI9z2cFpZJwHShKY2HE8HpaUrEtgsV3ydIZCZCaBw7rGDT84IAnsFRwU8QxeFTm5mPBW-0DjtW-NnF-heX8S-J9qOWhIrNw/s1600/il_570xN.508231534_91cr.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisyq4ZBhFlV4alTv_H3BJ-eYfXPtVgzSMHvhKa_3wD63gfI9z2cFpZJwHShKY2HE8HpaUrEtgsV3ydIZCZCaBw7rGDT84IAnsFRwU8QxeFTm5mPBW-0DjtW-NnF-heX8S-J9qOWhIrNw/s1600/il_570xN.508231534_91cr.jpg" height="213" width="320" /></a></div>
<div style="text-align: center;">
<span style="font-size: x-small;">Photo from <a href="https://www.etsy.com/it/listing/161548117/2pcs-set-infinito-ciclo-inknart">here</a></span></div>
<br />
<b>The lemniscate really exploded this year.</b> Even someone like me had to notice this trend in the skin of the people around. The most popular is probably the one with "love" embedded in it, or other writings (note the <a href="http://cantorontheshore.blogspot.it/2014/10/the-real-meaning-of-to-infinity-and.html">Buzz Lightyear</a> reminder!)<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjVontwNXKLEOZ1UhBbIwL-qItnhQ4lBeGjVY4vmf6Ecg1nO0XQyR5KotMB-YGC_6rVKcET7mNxVcQ51sZJ_iIwFuSFZAp8Sfmc-aCnTWUgzogeIiZLh5qWcQEb7vnd9ijlJxHlU9Snw/s1600/tumblr_mti0m0WAgR1szhlroo1_1280.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjVontwNXKLEOZ1UhBbIwL-qItnhQ4lBeGjVY4vmf6Ecg1nO0XQyR5KotMB-YGC_6rVKcET7mNxVcQ51sZJ_iIwFuSFZAp8Sfmc-aCnTWUgzogeIiZLh5qWcQEb7vnd9ijlJxHlU9Snw/s1600/tumblr_mti0m0WAgR1szhlroo1_1280.jpg" height="320" width="320" /></a></div>
<div style="text-align: center;">
<span style="font-size: x-small;">Photo from <a href="http://infinity-tattoos.net/page/16">here</a></span></div>
<br />
and the combos "feather+infinity" and "finger tattoo+infinity" are also ubiquitous.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjA03nmdPjzPWhb5IBArUQSC-UAXbJx8JnU3AbwxS4en4MvUIBxXOptUXllGgkGSJIjpgHxH2ViysiDtRQ9gcvq_NUh4G_ixO_i68JngWe1EeWE-qKeHNzXMrmwRL8y6hZCht2bFX5fXg/s1600/629dccdcabff86c6d24b4ca5b7faa005.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjA03nmdPjzPWhb5IBArUQSC-UAXbJx8JnU3AbwxS4en4MvUIBxXOptUXllGgkGSJIjpgHxH2ViysiDtRQ9gcvq_NUh4G_ixO_i68JngWe1EeWE-qKeHNzXMrmwRL8y6hZCht2bFX5fXg/s1600/629dccdcabff86c6d24b4ca5b7faa005.jpg" height="320" width="320" /></a></div>
<div style="text-align: center;">
<span style="font-size: x-small;">Photo from <a href="https://www.pinterest.com/pin/275001120968828712/">here</a></span></div>
<div style="text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTZixF7attTKa9z1wpiO8YofAL4H1mEXiIcAzLhxrRUBT8cXXclcOdZlMfKq85Z0ZGwW9oaonA5F5cC4lfAjQO6E3M1pJijBJ4_jd5iI51FVTrr_iShO1pnZHP5mKwIx0oikhHYBDj5g/s1600/infinity-tattoos--large-msg-134556696919.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTZixF7attTKa9z1wpiO8YofAL4H1mEXiIcAzLhxrRUBT8cXXclcOdZlMfKq85Z0ZGwW9oaonA5F5cC4lfAjQO6E3M1pJijBJ4_jd5iI51FVTrr_iShO1pnZHP5mKwIx0oikhHYBDj5g/s1600/infinity-tattoos--large-msg-134556696919.jpg" height="213" width="320" /></a></div>
<div style="text-align: center;">
<span style="font-size: x-small;">Photo from <a href="http://hannahrjones13.buzznet.com/photos/iswearweareinfinite/?id=68303501">here</a></span></div>
<br />
There are tumblrs devoted to <a href="http://infinity-tattoos.net/">it</a> and all <a href="http://www.cuded.com/2013/11/infinity-tattoo-ideas/">sorts</a> of <a href="http://weheartit.com/tag/infinity%20tattoo">websites</a>. But I am a set theorists, not a fashion designer, so what can I say to you about it?<br />
<br />
Well, now I am about to dump to you a nugget of information that will <i>surely</i> make you popular with your friends. You can ask: <b>why there is <i>one</i> symbol of infinity, when there are <i>many</i> infinities?</b> The reason is that <b>the lemniscate, in mathematics, always refers to the <i>potential</i> infinity</b>, i.e., the infinity you cannot reach, (I explain this <a href="http://cantorontheshore.blogspot.it/2014/10/the-real-meaning-of-to-infinity-and.html">here</a>) while<b> each different <i>actual</i> infinity, i.e., an infinity considered as size of an infinite set, has a different <a href="http://cantorontheshore.blogspot.it/2014/07/a-pint-of-aleph.html">symbol</a></b>. Cool, huh?<br />
<br />
<i>Pick up artist suggestion</i><br />
So, you are at a New Year party and you see a cute girl with an infinity tattoo. What do you do? You go close to her, you say "Hey, did you know that this symbol only means potential infinity? But the actual infinity..." Pause. "... is in my pants".<br />
<br />
And then you pick from your pants' pocket a piece of paper with an aleph on it. Mark my words, she'll throw herself to your arms. You're welcome.<br />
<br />
<i>You want a symbol that it's even more than infinity? <a href="http://cantorontheshore.blogspot.it/2014/08/infinity-times-infinity-updates.html">Here you are</a>. </i>Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com0tag:blogger.com,1999:blog-8155036766532593680.post-59442073887844403662014-12-22T06:20:00.000-08:002015-02-24T09:29:00.520-08:00Teoría de conjuntos (a poem for Christmas)<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9G0pcvlDgMPrQ2hPtbgy2attDmSi5RKJAMWAqh8ud52v3v8DOA97V8yCjXugbVwtklHv0iHh5phCE0UqZRkmJdaqHIwzGG9OdskeVbSMEkXuiTvYSShqIrpYm63LW3XotLoeCOIxiBg/s1600/Mario_Benedetti.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9G0pcvlDgMPrQ2hPtbgy2attDmSi5RKJAMWAqh8ud52v3v8DOA97V8yCjXugbVwtklHv0iHh5phCE0UqZRkmJdaqHIwzGG9OdskeVbSMEkXuiTvYSShqIrpYm63LW3XotLoeCOIxiBg/s1600/Mario_Benedetti.jpg" height="200" width="170" /></a></div>
<br />
Christmas is back! The most... depressing period of the year. Really, while the peak of suicides over Christmas is a <a href="http://www.cdc.gov/ViolencePrevention/suicide/holiday.html">myth</a>, it surely can bring the worst in our souls: extremely overcrowded shopping streets (with the included unnecessary touching, keep your heels far from my feet, woman, thanks), <a href="http://www.buzzfeed.com/briangalindo/17-santa-claus-photos-that-will-make-your-skin-crawl#.reRJ6n1Ok">creepy</a> Santa Claus(es?) grinning at every corner, sad, pale light decorations that always miss a row of burned bulbs and that in the most flamboyant cases will surely induce a <a href="https://www.youtube.com/watch?v=-VL9dQHMMfo">seizure</a>. And what about the <i>perfect</i> present you have to choose for your Nazi sympathizer cousin that you managed to avoid for one year?<br />
<br />
For this reason, this time I'll try to avoid the hard mathematics and I'll post something... warmer. A poem, whose title is "Set Theory"<br />
<br />
<div style="text-align: center;">
<b style="font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 18.1818180084229px;">Teoría de conjuntos</b><br />
<b style="font-family: Verdana, Arial, Helvetica, sans-serif; font-size: 18.1818180084229px;"><br /></b></div>
<div>
<div style="text-align: center;">
<span style="background-color: white; font-family: Verdana, sans-serif; font-size: 12px;">Cada cuerpo tiene</span></div>
<div style="text-align: center;">
<span style="background-color: white; font-family: Verdana, sans-serif; font-size: 12px;"><span style="font-size: 12px;">su armonía y</span></span></div>
<span style="background-color: white; font-family: Verdana, sans-serif; font-size: 12px;">
</span><span style="background-color: white; font-family: Verdana, sans-serif; font-size: 12px;"></span>
<div style="text-align: center;">
<span style="background-color: white; font-family: Verdana, sans-serif; font-size: 12px;"><span style="font-size: 12px;">su desarmonía.</span></span></div>
<span style="background-color: white; font-family: Verdana, sans-serif; font-size: 12px;">
</span><span style="background-color: white; font-family: Verdana, sans-serif; font-size: 12px;"><div style="text-align: center;">
<span style="font-size: 12px;">En algunos casos</span></div>
</span><span style="background-color: white; font-family: Verdana, sans-serif; font-size: 12px;"><div style="text-align: center;">
<span style="font-size: 12px;">la suma de armonías</span></div>
</span><span style="background-color: white; font-family: Verdana, sans-serif; font-size: 12px;"><div style="text-align: center;">
<span style="font-size: 12px;">puede ser casi</span></div>
</span><span style="background-color: white; font-family: Verdana, sans-serif; font-size: 12px;"><div style="text-align: center;">
<span style="font-size: 12px;">empalagosa.</span></div>
</span><span style="background-color: white; font-family: Verdana, sans-serif; font-size: 12px;"><div style="text-align: center;">
<span style="font-size: 12px;">En otros</span></div>
</span><span style="background-color: white; font-family: Verdana, sans-serif; font-size: 12px;"><div style="text-align: center;">
<span style="font-size: 12px;">el conjunto</span></div>
</span><span style="background-color: white; font-family: Verdana, sans-serif; font-size: 12px;"><div style="text-align: center;">
<span style="font-size: 12px;">de desarmonías</span></div>
</span><span style="background-color: white; font-family: Verdana, sans-serif; font-size: 12px;"><div style="text-align: center;">
<span style="font-size: 12px;">produce algo mejor</span></div>
</span><span style="background-color: white; font-family: Verdana, sans-serif; font-size: 12px;"><div style="text-align: center;">
<span style="font-size: 12px;">que la belleza.</span></div>
</span><span style="font-family: Verdana, sans-serif; font-size: 12px; margin: 0px; padding: 0px;"><br style="margin: 0px; padding: 0px;" /></span></div>
<div style="text-align: center;">
<span style="font-size: x-small;"> (taken from <a href="http://www.poemas-del-alma.com/mario-benedetti-teoria-de-conjuntos.htm">here</a>)</span></div>
<div style="text-align: center;">
<br /></div>
<div style="text-align: left;">
This is a poem by <a href="http://en.wikipedia.org/wiki/Mario_Benedetti">Mario Benedetti</a>, a Uruguyan poet, probably one of the most important writers in Latin America in the last century, even if he is not really known in the English-speaking world. </div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
Unfortunately, my knowledge of Spanish and English is not enough for a good translation, so if you don't know Spanish (hello, 93% of the world population!), I can just give you the meaning. In the end, it has nothing to do with maths: "conjunto" in English is "set", but also "collection", "whole". Benedetti says that every body has a harmony and a disharmony. somethimes the sum of harmonies can be almost loathsome, while in other cases the collection of disharmonies can produce something even better than beauty.</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
So, nothing to say about math here, the "set theory" of the title was just the inspiration for Benedetti to write a small and wise poem about the beauty of the imperfection of human nature. Keep it with you this days: what if the collection of all the horrors we will have to endure for the holidays, in the end really forms something beautiful?</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
I'll take Benedetti's word here.</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
<i>Thanks to Luz for having pointed me out this poem! See? The suggestions box works!</i></div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
<i>Looking for something light? Then <a href="http://cantorontheshore.blogspot.com/2014/08/dont-be-naive-google.html">naive Set Theory</a> is for you. </i></div>
Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com0tag:blogger.com,1999:blog-8155036766532593680.post-2896858331780578292014-11-03T10:48:00.001-08:002017-06-20T06:00:52.757-07:00IntermissionIt is time for this blog to take a little pause. I mean, you wouldn't think that finding set theory instances in pop culture would grant the same rapid fire coverage of, say, <a href="http://www.kittyhell.com/">Hello Kitty Hell</a>, would you? <br />
<br />
So, today no pop culture. Instead here is a video:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<iframe width="320" height="266" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/KDCJZ81PwVM/0.jpg" src="https://www.youtube.com/embed/KDCJZ81PwVM?feature=player_embedded" frameborder="0" allowfullscreen></iframe></div>
<br />
<br />
This is the Full Program: Infinity part of the World Science Festival. A Philosopher/Theologician, a Mathematician, a Set Theorists and a Physicist meet together, and instead of starting a joke they discuss for more than one hour about infinity. If you are seriously interested in infinity, this is a nice video to be introduced to, with different perspectives. In parts it is a bit technical, so take your time, don't sweat it.<br />
<br />
The introduction is fantastic, with a parade of people talking about infinity in absurdely vague and personal terms, all thinking that infinity is central in their work, like a yoga man, a librarian (!) and of course, an ultrafinitist (that, frankly, doesn't really come out well from the comparison).<br />
<br />
Sit back and enjoy the ride!<br />
<br />
<i><a href="http://cantorontheshore.blogspot.com/2014/09/god-and-big-bang-theory.html">Here</a> is some more connection between Set Theory and Theology. Oh, and the above is divulgation done right. Wanna see <a href="http://cantorontheshore.blogspot.com/2015/01/infinite-mess-part-one.html">divulgation done wrong</a>?</i>Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com0tag:blogger.com,1999:blog-8155036766532593680.post-25731298937711942112014-10-21T00:37:00.000-07:002015-02-24T09:14:46.553-08:00The Real Meaning of To Infinity and Beyond!Yes, I went there. I finally acknowledged the elephant in the room, and I am writing now about the most used and abused quote about infinity.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEixtDuihe8SiwGMC3JSQoxVynuKgESbUV0_3UWuoYwlijeK3lA_wKQbWxEqyomb5goRiDU3utrVx-v9ezRdLC8P-lwxgiQYJHl56CpRNB8SVnjvIfHa2H9EekQxl-hEcs74B8aoiKudgg/s1600/Blast-Off-Buzz-Lightyear.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEixtDuihe8SiwGMC3JSQoxVynuKgESbUV0_3UWuoYwlijeK3lA_wKQbWxEqyomb5goRiDU3utrVx-v9ezRdLC8P-lwxgiQYJHl56CpRNB8SVnjvIfHa2H9EekQxl-hEcs74B8aoiKudgg/s1600/Blast-Off-Buzz-Lightyear.jpg" height="263" width="320" /></a></div>
<div style="text-align: center;">
<span style="font-size: x-small;">To... boundlessness and... above?</span></div>
<div style="text-align: center;">
<br /></div>
<div style="text-align: left;">
I'm talking of course about the infamous catchphrase of Buzz Lightyear in Toy Story or, as Imdb puts it:</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
<span style="background-color: white; color: #333333; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 17.0300006866455px;">[</span><i style="background-color: white; color: #333333; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 17.0300006866455px;">repeated line</i><span style="background-color: white; color: #333333; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 17.0300006866455px;">] </span><br />
<i style="background-color: white; color: #333333; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 17.0300006866455px;"><a href="http://www.imdb.com/name/nm0000741/" style="color: #70579d; text-decoration: none;">Buzz</a></i><span style="background-color: white; color: #333333; font-family: Arial, Helvetica, sans-serif; font-size: 13px; line-height: 17.0300006866455px;">: To infinity, and beyond! </span></div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
<b>What is the meaning of "To infinity and beyond?"</b> </div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
It seems a throwaway line, but hidden there is a <b>deep philosophical meaning</b>. It pops up here and there in pop culture and even in random discussions with friends. In <a href="http://en.wikipedia.org/wiki/2001:_A_Space_Odyssey_(film)">2001: A Space Odissey</a> just before the psychedelic "Star Gate" sequence a title card appear, "Jupiter and Beyond the Infinite" (an inspiration for Buzz?). In the hit <a href="https://www.youtube.com/watch?v=4m1EFMoRFvY">Single Ladies (Put a Ring on It)</a>, Beyoncé sings "... and delivers me to a destiny, to infinity and beyond.", There are then the album <a href="http://en.wikipedia.org/wiki/Beyond_the_Infinite">Beyond the Infinite</a>, by trance group Juno Reactor, or the Greek film <a href="http://en.wikipedia.org/wiki/Eternity_and_a_Day">Eternity and a Day</a>, by director Theo Angelopoulos. In 2008 this sentence helped father and son <a href="http://www.foxnews.com/story/2008/09/10/toy-story-line-helped-father-son-survive-in-water-for-15-hours/">to survive a shipwreck</a>. And let's not forget <a href="http://cantorontheshore.blogspot.co.at/2014/08/advanced-thinking-thinking-infinity.html">infinity plus one</a>! And the lovey-dovey couples smooching "I love you infinity times and more". There are probably infinite other examples (and more!). </div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
It sounds paradoxical: <b>how can you go <i>beyond</i> the infinity</b>? It's not possible to reach the end of infinity (because it is infinite), so how can one overtake it? One needs to change completely point of view: from <i>potential</i> infinity to <a href="http://en.wikipedia.org/wiki/Actual_infinity">actual infinity</a>.<br />
<br />
Let me go back to the example of <a href="http://cantorontheshore.blogspot.co.at/2014/07/the-fault-in-fault-in-our-stars-part-1.html">Achilles and the tortoise</a>, just like this blog started. You have the tortoise and Achilles, say, 10 meters apart, Achilles in the back. They run. Achilles is much faster than the tortoise, so when Achilles reached the point where the tortoise was, the tortoise just made one meter. Achilles run that meter, and the tortoise is 10 cm ahead. And so on, and so on. Will Achilles ever reach the tortoise?<br />
<br />
If you have the mentality above, it will not: you have this infinite succession of states: 10 m distance, 1 m distance, 10 cm distance, ... and you cannot reach the <i>end</i> of it, because it is infinite! This is the so-called potential infinite, potential because you never realize it in full, you just see parts of it.<br />
<br />
But we know that Achilles reaches the tortoise, right? We're not that stupid. So what? Well, this is where Cantor shines! He managed to treat in a formal way the <i>actual</i> infinite. <b>You pick all the infinity, you put it in a sack, and you treat it as a whole, <i>completed</i> object.</b> So you can imagine what happens after it, in this case Achilles overtake the tortoise, turns and makes very funny faces. "Brlbrblrblrl".<br />
<br />
Think of the potential infinity as an infinity you are walking inside. Like, just go out now and try to reach the horizon, it will always go on and on and you will never reach it. But the actual infinity is like seeing everything from above, so instead of walking pick a drone and go up, up, up, and <b>you can embrace everything with a single view</b>. (Of course, the metaphor works only if our Earth is <a href="http://theflatearthsociety.org/cms/">flat</a> and infinite, sorry, so I guess that makes no sense).<br />
<br />
Another Zeno's Paradox: the dichotomy paradox. You have to go to a bus station that is 1 km far. To go there, you have to go half kilometer far. And then a quarter kilometer more. And then an eighth more. How can you reach the bus station? In mathematics, the distance you make is written like this:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgY-q7W5a951IDqTU9cZnoYzUQunGfDZ5_soHjip4WIKMH2N89sxD0S7MFcYAyoavZbbVi3iVaSFfYCdFaT_3UfmDSBEjE9jyrPBmKsfEYpGVWPtVcWnSaDFIMWzJHH6WR5EnjGWjce3A/s1600/dichotomy.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgY-q7W5a951IDqTU9cZnoYzUQunGfDZ5_soHjip4WIKMH2N89sxD0S7MFcYAyoavZbbVi3iVaSFfYCdFaT_3UfmDSBEjE9jyrPBmKsfEYpGVWPtVcWnSaDFIMWzJHH6WR5EnjGWjce3A/s1600/dichotomy.png" /></a></div>
According to a potentialist, this makes a potential 1, that is, is closer and closer to 1 without ever reaching it. According to an actualist, this is 1. What is the difference, you say? Look at this:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilzFJfaawQAfJEkYpJ1m_iaxqyHrjgS7U5ZZnN2dzbA11iEePhUBhjoT3e4L3PasqYXyfhLyv2UfkYBEfl9-UanBESHl92awPrxf__XCfNXUKjGL_GDlpfKALvVpl6GJ5VXeSoG50Lhw/s1600/Dichotomy+2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilzFJfaawQAfJEkYpJ1m_iaxqyHrjgS7U5ZZnN2dzbA11iEePhUBhjoT3e4L3PasqYXyfhLyv2UfkYBEfl9-UanBESHl92awPrxf__XCfNXUKjGL_GDlpfKALvVpl6GJ5VXeSoG50Lhw/s1600/Dichotomy+2.png" /></a></div>
How much is this? According to an actualist, it is 2. According to a potentialist, it is BLAARGH GET THAT AWAY FROM MY FACE! SATAN!<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhS0xuSTg78XfQvkwgoReNLzd5Fs1WXUSFtARhadLtaWj6SbPl2AM975KdsmgGQJR0XcrdTtShsBO_aGsPowGvnGb9XtpgXvtyUO5ivCPsAnH-EcGhKtA2P-eoOBmUZsNos0G6EDaeGCA/s1600/PanicKronecker.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhS0xuSTg78XfQvkwgoReNLzd5Fs1WXUSFtARhadLtaWj6SbPl2AM975KdsmgGQJR0XcrdTtShsBO_aGsPowGvnGb9XtpgXvtyUO5ivCPsAnH-EcGhKtA2P-eoOBmUZsNos0G6EDaeGCA/s1600/PanicKronecker.gif" height="180" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div style="text-align: center;">
<span style="font-size: x-small;">Artist's rendition</span></div>
</div>
<div style="text-align: left;">
<br />
The potentialists say that this is just a trick, that our limited minds just cannot comprehend actual infinities, and yet, mathematics goes just smoother with them, and the paradoxes are solved. Thanks, Cantor!<br />
<br />
You know what? Let me see what the Internet thinks that "To infinity and beyond" means...<br />
<br />
Oh, boy.<br />
<br />
Fun fact: there are many, many mathematicians, even big experts, even Fields medals, that just have no clue about this. They work in mathematics like they were in the 19th century, willfully ignoring that all their work is founded on this.<br />
<br />
This is the first results in Google (hopefully the second after I post this).<br />
<br />
<span style="background-color: #f7f7f7; color: #545454; font-family: Arial; font-size: 14px; line-height: 21.6000003814697px;">Definition in Mathematical Circles:</span><br />
<span style="background-color: #f7f7f7; color: #545454; font-family: Arial; font-size: 14px; line-height: 21.6000003814697px;">What exactly does Buzz Lightyear mean when he says, "To Infinity and beyond!"? A few professionals at Harvard investigated the origin of this quote and traced it back to limits. According to Dr. Sanjay Gupta, "Buzz Lightyear is a metaphor of a function which approaches a certain number, but never actually reaches it." However, experts at MIT believe that Buzz Lightyear is referring to vertical asymptotes. Dr. Benjamin Hernandez says, "It is possible to cross horizontal asymptotes, but verticals are impossible. Buzz Lightyear is showing everyone that he can do the impossible and cross horizontal and vertical asymptotes."</span><br />
<br /></div>
<div style="text-align: left;">
Now, I don't know who are these guys. Most probably they are misquoted, maybe they don't exists, so I am not really against them. But what is written there, trying to sound professional as heck (Harvard! MIT! Bum!) is far-reaching or just nonsense. The first sentence, by Dt Sanjay Gupta, is a typical expression of potential infinity (it approaches a certain number but never actually reaches it) and does not explain anything. Where is the beyond? The second sentence makes more sense: here is a horizontal asymptotes:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEggFZqzBd3QzvbZuwTOeH6i_zNhAQS3Q-mS9xYBiZjIIyFktF7_9khySnf4h7YrCAG0AyhUo2_2tCWoFzTIvgGxuf03QCTwqOEMBUfWELSA3NzHryz4_Fv6YW5nDylho4telCvUPqavlg/s1600/Limits_at_Infinity_03.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEggFZqzBd3QzvbZuwTOeH6i_zNhAQS3Q-mS9xYBiZjIIyFktF7_9khySnf4h7YrCAG0AyhUo2_2tCWoFzTIvgGxuf03QCTwqOEMBUfWELSA3NzHryz4_Fv6YW5nDylho4telCvUPqavlg/s1600/Limits_at_Infinity_03.gif" height="156" width="320" /></a></div>
<br />
The asymptote is that dashed line. See? The function crosses it. Here is a vertical asymptote:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYSdrrVbPG1qULexzxJfZHw8qe85ZhgzgsY4pL8DRIsC_CQDFfbJpkfTFPKYEeM61lRi6m-irOAc-MkT4M-5-GVdYyvR8VgIJy6hzjP4Wcq7X5dkUsHw3tyIzRIXtBL6bBAiIfxqBQXw/s1600/asymptote_vertical.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYSdrrVbPG1qULexzxJfZHw8qe85ZhgzgsY4pL8DRIsC_CQDFfbJpkfTFPKYEeM61lRi6m-irOAc-MkT4M-5-GVdYyvR8VgIJy6hzjP4Wcq7X5dkUsHw3tyIzRIXtBL6bBAiIfxqBQXw/s1600/asymptote_vertical.gif" /></a></div>
The function doesn't cross the line. And Buzz <i>can</i>! Well, OK, but... isn't this a little bit too technical? Also, isn't it a bit underwhelming that "Infinity and beyond" means a little bump in a line, just like your body when you eat too much Marshmallow Fluff?<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2kOwwkdZ4V5uC9E0fY1XoTzYDonpHzLJeQFFTFzkYd1_0yN4_8W24SMrcpVPj7UkupB8_kdl57Jv43ECxmno3ecJ0gXHfrHnk3Lpq7UX_K22sw1u3Sk8qh24uTqPCvsnDfHuBzk6Zyw/s1600/asymptote_vertical+2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2kOwwkdZ4V5uC9E0fY1XoTzYDonpHzLJeQFFTFzkYd1_0yN4_8W24SMrcpVPj7UkupB8_kdl57Jv43ECxmno3ecJ0gXHfrHnk3Lpq7UX_K22sw1u3Sk8qh24uTqPCvsnDfHuBzk6Zyw/s1600/asymptote_vertical+2.png" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<span style="text-align: left;"><span style="font-family: inherit; font-size: x-small;">INFINITY!</span></span></div>
<div class="separator" style="clear: both; text-align: center;">
<span style="text-align: left;"><span style="font-family: inherit; font-size: x-small;"><br /></span></span></div>
<div class="separator" style="clear: both; text-align: left;">
So, forget about what you read online. The meaning of "To infinity and beyond", even mathematically, is: <b>we all think that we are trapped in our human limits, without escape, but in the end it is just an illusion. Buzz (and Cantor) is showing us the way to recognize the illusion, change the perspective, finally break free and go! leaving all our chains behind, going were it was previously unthinkable, unimaginable</b>.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjnW3EIhsdCZkFVvy1GIZp-9ZtO3nJIPTZLbpMpaD84GVVSSFBApmjydSx7MfBYiKus0OEoJhF-MZGk3RoWsG9yVVdPYubOxQ8Bbnm0p_itLiMHH26kV2jZ0tfQWU84FRowIAObqgIoSw/s1600/Buzz+Cantor.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjnW3EIhsdCZkFVvy1GIZp-9ZtO3nJIPTZLbpMpaD84GVVSSFBApmjydSx7MfBYiKus0OEoJhF-MZGk3RoWsG9yVVdPYubOxQ8Bbnm0p_itLiMHH26kV2jZ0tfQWU84FRowIAObqgIoSw/s1600/Buzz+Cantor.png" height="263" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
You hear that, potentialists?</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
PS. I am actually going to Harvard next week. If I meet Dr. Sanjay Gupta, I'll let you know.</div>
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
<i>Statistics says that probably this is the first page of the blog you are reading. Then, may I suggest to read the <a href="http://cantorontheshore.blogspot.com/2014/07/this-blog-will-blow-your-mind-aka.html">manifesto</a>, to understand what is this about? Or just skip to the meaty parts, like the posts on "<a href="http://cantorontheshore.blogspot.com/2014/07/the-fault-in-our-infinites-part-2.html">The Fault in Our Stars</a>", or "<a href="http://cantorontheshore.blogspot.com/2014/09/god-and-big-bang-theory.html">The Big Bang Theory</a>", or a cute <a href="http://cantorontheshore.blogspot.com/2014/08/advanced-thinking-thinking-infinity.html">AT&T advertisement</a>.</i></div>
</div>
Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com9tag:blogger.com,1999:blog-8155036766532593680.post-62845283467112066212014-10-06T08:48:00.001-07:002019-11-24T01:13:25.872-08:00One Reinhardt and counting...<span style="font-size: x-small;"><i>The original post has been changed: the attribution of the game has been clarified.</i></span><br />
<br />
Can you hear it? Do you hear that sound? It's something like <i>streeeeeeeeaaaaaaatch</i>. It's me stretching the theme of the blog to accomodate this topic.<br />
<br />
You know <a href="http://en.wikipedia.org/wiki/2048_%28video_game%29">2048</a>? Of course, everybody knows it. It's a very addicting game by <a href="http://gabrielecirulli.com/">Gabriele Cirulli</a>, and the Wall Street Journal called it "almost Candy Crush for math geeks".<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgT9WjtPPf1RslX3tjhzfS7Q02xPAvDJ1matL8QCyGO90R6aSURBfpeIJOdOFf10MsywBcCpcxXNaIxWQantZR_nPG0krjY6QDke9aZTR_LaV1AsYDL6bQzG-M682ZnhboJvrHgelkAkA/s1600/2048_Screenshot.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgT9WjtPPf1RslX3tjhzfS7Q02xPAvDJ1matL8QCyGO90R6aSURBfpeIJOdOFf10MsywBcCpcxXNaIxWQantZR_nPG0krjY6QDke9aZTR_LaV1AsYDL6bQzG-M682ZnhboJvrHgelkAkA/s1600/2048_Screenshot.png" width="319" /></a></div>
<div style="text-align: center;">
<span style="font-size: x-small;">Addiction has a face.</span></div>
<br />
I cannot describe it. Just <a href="http://gabrielecirulli.github.io/2048/">play</a> it. Or don't! You have to read this blog, don't get distracted.<br />
<br />
The game is <a href="https://github.com/gabrielecirulli/2048">open source</a>, so anybody can make a personal version. But then, don't you feel that the numbers here are a bit... small? 2048 is so tiny, is there a way to go really big?<br />
<br />
If you dare to adventure the deep, deep academia web you'll find a specific version... with <a href="http://cantorontheshore.blogspot.co.at/2014/09/artemis-fowl-and-large-cardinals_22.html">large cardinals</a>! <b>Now also you can climb up the large cardinals hierarchy</b>! The name of the game is <a href="https://zhuyizheng.blog/2048.htm">Reinhardt</a>. The original idea was by Yizheng, and then it was picked up by Chris La Sueur, that changed the large cardinals and published it.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjErnb2uuJ4VbXcNWwGZfiHJzpsciKIG0yw1Yg0NI_ZTnkMCs8DJ9elQzhr55wHxHKJDRr6YXQooXpyjdpCWU-m3ijoNAVji1A15pPo2BFADbplEN7VxYx3axu2Iv9TkS_taKfY6ZymRA/s1600/Reinhardt.png" width="319" /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBdLUW68h0OpOTnnw65Btd-aKODhsAGvXyT5uxICG7a0IKwJMcx2q9LAj9A-X_Niil3EgCR6vs9fa1XYG_OczeKii2Uabcv2rEOeGoSIWL7qkbHdtKaJbuVHPGN4NPMIyfJG6ylGx_fg/s1600/Reinhardt.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><br /></a></div>
Now, Chris is so kind to suggest you not to play to the big version, but I am evil and I push you to try that instead of Reinhardt. It is called <a href="http://www.maths.bris.ac.uk/~cl7907/reinhardt/131072.htm">1=0</a>, and it is cleaner and more strategical.<br />
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-8CALwdF3JpsEoOsD-eLhiEgi03KAMx8eIaKjHSMgj2Ck0x-h77Lgz6h88fAFMAuG3knkAG3GbiZ3QUkJmsuue1sRB734EpvEouWHWtUHlImeRZJRGSUJ6Y_cdOmfPRAGNBLvThUK2g/s1600/1=0.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-8CALwdF3JpsEoOsD-eLhiEgi03KAMx8eIaKjHSMgj2Ck0x-h77Lgz6h88fAFMAuG3knkAG3GbiZ3QUkJmsuue1sRB734EpvEouWHWtUHlImeRZJRGSUJ6Y_cdOmfPRAGNBLvThUK2g/s1600/1=0.png" width="319" /></a></div>
<br />
I thought about giving you the list of the cardinals involved, but why ruining the fun? Part of the delight is in the exploration!<br />
<br />
But why it is called Reinhardt? This is the skeleton in the closet of all set theorists.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-vg0rcfa5_eNVy0qCBlbnRv3JRz78HVm53yj5yjTOKJPAIEkxL1mvqP2Ka2KafkVRpKO_k6DmqSKXl3e1R2jN4_bqvz6cm7OUn1LnPpyXe90TYCNhpV3sp2aJM64VxIlNInFgAihu6g/s1600/faces+of+set+theory.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="250" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-vg0rcfa5_eNVy0qCBlbnRv3JRz78HVm53yj5yjTOKJPAIEkxL1mvqP2Ka2KafkVRpKO_k6DmqSKXl3e1R2jN4_bqvz6cm7OUn1LnPpyXe90TYCNhpV3sp2aJM64VxIlNInFgAihu6g/s400/faces+of+set+theory.png" width="400" /></a></div>
<br />
<div style="text-align: center;">
<span style="font-size: x-small;">The reaction of set theorists when asked about Reinhardt and large cardinals</span></div>
<br />
<br />
Once upon time (1969, to be precise), William Reinhardt was studying some large cardinal, and had an idea: <b>to build the largest cardinal ever</b>, the king of large cardinals, the most powerful! We now call it <a href="http://en.wikipedia.org/wiki/Reinhardt_cardinal">Reinhardt cardinal</a>.<br />
<br />
But just after months, <a href="http://en.wikipedia.org/wiki/Kenneth_Kunen">Kenneth Kunen</a> proved that it was too big and powerful. In fact <b>it just couldn't exist</b>! Set theorists wanted the absolute power, but they overshot and reached a contradiction, and now the tale of the non-existing Reinhardt cardinal is taught to all students, as a sombre reminder that in trying to achieve too much, one risks to destroy everything. Anyway, <b>Reinhardt cardinal is the largest cardinal, even if it does not exists</b>, and the aim of the game is to reach it.<br />
<br />
And why the big one is called 1=0? This is more difficult to say, probably I will write it in another post. (<i>Edit: <a href="http://cantorontheshore.blogspot.it/2015/06/the-truth-about-zero-theorem.html">here it is</a>!)</i><br />
<br />
Now you can go playing! Have a nice waste of time!<br />
<br />
<i>Many large cardinals that appear in the game already appeared <a href="http://cantorontheshore.blogspot.com/2014/09/artemis-fowl-and-large-cardinals.html">here</a>. Interested in just one large cardinal? Make it <a href="http://cantorontheshore.blogspot.com/2014/09/god-and-big-bang-theory.html">this one</a>, says Sheldon.</i><br />
<br />
<br />Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com11tag:blogger.com,1999:blog-8155036766532593680.post-53327904562645644152014-09-29T07:17:00.004-07:002016-07-06T05:48:08.526-07:00God and the Big Bang TheoryThe title caught your attention? Good! It would seem at the beginning that this has nothing to do with set theory, but trust me and go on.<br />
<br />
This time I am going to talk about the Big Bang Theory. Yes, not the awe-inducing cosmological model for the early development of the universe, but the sitcom (notice the uppercase T?). What can I say about this sitcom that has never been said before? Extremely popular, it lures the nerds with tons of citations about science, science fiction and fantasy, but at the same time it ridiculizes them, but anyway less than other series in the past. The humour is predictable and comforting. Anyway, everybody agrees that, like all the TV series, with time it became less interesting, so I stopped seeing it.<br />
<br />
Until this week! When <a href="http://math.stackexchange.com/questions/942581/big-bang-theory-reference-to-formal-logic">this discussion</a> on Mathematics Stack Exchange made me curious again, and I have seen the episode 8x02 "The Junior Professor Solution", aired the 22 September 2014. So, first of all, kudos to Doug Spoonwood for having spotted this, and to Asaf Karagila for being the first to realize its significance. I cannot link the video, of course, so I'll describe the scene. Sheldon, the genius of the series, has to prepare a lesson for Wolowitz, and he wanted it to be as hard as possible. Sheldon is showing whiteboards full of difficult formulas, saying<br />
<br />
<span style="font-family: "courier new" , "courier" , monospace;">Oh, I'm working on my lesson plan for Wolowitz. He is going to be so lost. Look at this section over here.</span><br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaVMuufUQf_wOEhj0qSzd8t4Lfc7ZZP7GMP6_DZG7J4P2cjVxoNX1RXKKki4REPiqxTBxKRF6Sg1Yw3MC5ZlVG2AxYULxjN_mrZQScCmcFcxQEGB0GXn6IezNjn4xeHl2_OR0_6EpXNg/s1600/Big+Bang+Theory.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="179" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaVMuufUQf_wOEhj0qSzd8t4Lfc7ZZP7GMP6_DZG7J4P2cjVxoNX1RXKKki4REPiqxTBxKRF6Sg1Yw3MC5ZlVG2AxYULxjN_mrZQScCmcFcxQEGB0GXn6IezNjn4xeHl2_OR0_6EpXNg/s1600/Big+Bang+Theory.jpg" width="320" /></a></div>
<br />
<span style="font-family: "courier new" , "courier" , monospace;">Even I don't really understand it.</span><br />
<span style="font-family: "courier new" , "courier" , monospace;"><br /></span>
<span style="font-family: inherit;">I DO! I DO! You suck, Sheldon! Ha! Let's see it closer:</span><br />
<span style="font-family: inherit;"><br /></span>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2APHf8yxJAWL7YMT1ibfQv5vdX8FMLw1wo_99W_bJRF1VAC7DewFHV48KnJHsqnPPWGY3x-dfBLaN50rTA5kZghVUoYFEknx9xE6hW2SneO336tdsh0Fka6OD0JaINPyXQlRfiG-c3g/s1600/God.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2APHf8yxJAWL7YMT1ibfQv5vdX8FMLw1wo_99W_bJRF1VAC7DewFHV48KnJHsqnPPWGY3x-dfBLaN50rTA5kZghVUoYFEknx9xE6hW2SneO336tdsh0Fka6OD0JaINPyXQlRfiG-c3g/s1600/God.png" /></a></div>
<span style="font-family: inherit;"><br /></span>
<span style="font-family: inherit;">This is <b>the famous proof by </b></span><b>Gödel of the existence of God</b> (better known as <a href="http://en.wikipedia.org/wiki/G%C3%B6del%27s_ontological_proof">Gödel's ontological proof</a>). Woah. I will leave you a moment to contemplate the magnificence of this discovery.<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
No, no, no, what are you thinking? Militant atheists, don't think that Gödel was a crackpot (well, he was, but after that). Dawkins, you got it wrong (not for the first time)! And militant religious, don't go around saying that the existence of God is <i>logical </i>(gah). Why do you always have to fight? He didn't <i>really</i> proved the existence of God, in reality. The right way to see it is as part of the history of philosophical logic.<br />
<br />
It was pretty common in the old days to try <a href="http://en.wikipedia.org/wiki/Ontological_argument">to prove the existence of God</a> <b>via logical arguments</b>. Think Anselm of Canterbury, Descartes, guys like that. <b>Gödel found logical mistakes in those arguments, and just rewrote them in the current logic language</b>, i.e. <a href="http://en.wikipedia.org/wiki/Modal_logic">modal logic</a>, which distinguishes between necessary truths (something that must be true, no matter what) and contingent truths (something that is true, but just because, if it weren't true everybody would be chill). Nothing really new, then, and nothing <i>real</i>. <b>It is just syntax, empty words, he was interested in the reasoning, not in its connections with reality.</b><br />
<br />
Basically (I am simplifying here) Gödel said:<br />
<br />
A1) a <b>positive</b> property can have only positive properties as consequences (optimist)<br />
A2) a property is <b>positive</b> if and only if its negation is not positive<br />
A3) being God is <b>positive </b>(this is slightly an understatement)<br />
A4) if a property is <b>positive</b>, it is because it must be positive (nothing left to chance here)<br />
A5) necessary existence is a <b>positive</b> property (again, quite optimist, if something must exists, then it is good)<br />
<br />
<i>If</i> this can be, then there exists an object with all the positive properties, i.e., God. That's it. All the details are in the Wikipedia link, it's pretty formal, but it is not difficult. He could have written "fairy" or "vegetable-y" instead of "positive", and "unicorn" or "carrot" instead of "God", and he would have proved the existence of a unicorn or of a carrot.<br />
<br />
But look again at Sheldon's whiteboard! See, there are A1, A2 etc also there. I did it on purpose, what is written in the whiteboard correspond exactly to what I wrote here. Almost. There is a horrible mistake.<br />
<br />
This is what Sheldon wrote, translated from formalese to English:<br />
<br />
A2) a property is positive if and only if its negation is positive<br />
<br />
But Sheldon! What did you write?! How can this be? You really didn't understand the proof.<br />
<br />
And what about Set Theory? This is where things got interesting. I told you that God exists <i>if</i> there exist positive properties. Do they? <b>Harvey Friedman has a <a href="http://u.osu.edu/friedman.8/files/2014/01/ConMathThe122512-1iea8ps.pdf">manuscript</a></b> on that, where he defines God and positive properties more carefully, and proves that <b>they exist if</b>... if...<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUS7bF3_908zBW1J-T_M1B2NGlkhQIMZxBwA0tTcj1y1X75veGelHeOsvDw5Wdy5slAGiFe5g0hRna4zWWTlSjjSXsrYZqPi29uT0sd6alIorJep-rDRIk13QOW7GZH69Qqk-qy6BNxQ/s1600/Measurable+cardinal.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="200" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUS7bF3_908zBW1J-T_M1B2NGlkhQIMZxBwA0tTcj1y1X75veGelHeOsvDw5Wdy5slAGiFe5g0hRna4zWWTlSjjSXsrYZqPi29uT0sd6alIorJep-rDRIk13QOW7GZH69Qqk-qy6BNxQ/s1600/Measurable+cardinal.png" width="320" /></a></div>
<div style="text-align: center;">
<i>Ta-da!</i></div>
<br />
<b>if there exists a <a href="http://en.wikipedia.org/wiki/Measurable_cardinal">measurable cardina</a>l</b>! Step aside, <i>Pope</i>. We did your job. Talking about <a href="http://cantorontheshore.blogspot.co.at/2014/09/artemis-fowl-and-large-cardinals_22.html">solving unsolvable problems with large cardinals</a>!<br />
<br />
Update: it turns out that Gödel hit a wall here. In 2013 <a href="http://page.mi.fu-berlin.de/cbenzmueller/papers/C55.pdf">someone managed to prove</a> (with Artificial Intelligence, nonetheless!) that the hypothesis A1-A5 are actually contradictory taken all together, so Gödel proof it's just wrong. Alas, all this post for nothing.<br />
<br />
<i>You want to build a measurable y yourself? Go <a href="http://cantorontheshore.blogspot.it/2014/10/one-reinhardt-and-counting.html">here</a>. You like to see discussions about infinity between theologians and set theorists? Then <a href="http://cantorontheshore.blogspot.com/2014/11/intermission.html">this page</a> is just for you.</i><br />
<span style="font-family: inherit;"><br /></span>
Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com0tag:blogger.com,1999:blog-8155036766532593680.post-4027428239243678662014-09-22T05:32:00.000-07:002015-02-24T08:57:57.567-08:00Artemis Fowl and the Large Cardinals Paradox (part two)<i>This post continues from <a href="http://cantorontheshore.blogspot.co.at/2014/09/artemis-fowl-and-large-cardinals.html">here</a>.</i><br />
<br />
Thanks for waiting. So, we were talking about this:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijfqP0v1haN3wbSaCHTsVdVLfpKhMUqK5po1ENmkr-5qouCmRH5GTLo9xZbTFOe2faERbwPPY5Bzxl-9E6_ZkHN8c2j-Hgb4eyv-r6g0Yba6ObtYf5et5W7gPDa6mLsGJ0vq50cHyLEg/s1600/ArtemisFowlTimeParadoxCover.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijfqP0v1haN3wbSaCHTsVdVLfpKhMUqK5po1ENmkr-5qouCmRH5GTLo9xZbTFOe2faERbwPPY5Bzxl-9E6_ZkHN8c2j-Hgb4eyv-r6g0Yba6ObtYf5et5W7gPDa6mLsGJ0vq50cHyLEg/s1600/ArtemisFowlTimeParadoxCover.gif" height="320" width="210" /></a></div>
<br />
this sentence in particular:<br />
<br />
<span style="font-family: "Courier New",Courier,monospace;"><span style="background-color: white; color: #666666; font-size: 13.63636302948px; line-height: 16.7999992370605px;">He focused on the high end of his intellect, solving unprovable theorems with large cardinals and composing an ending for Schubert's unfinished Symphony N° 8.</span></span><br />
<br />
and we were trying to understand what it means "<b>solving unprovable theorems with large cardinals</b>". In the previous post I introduced <a href="http://cantorontheshore.blogspot.co.at/2014/09/artemis-fowl-and-large-cardinals.html">large cardinals</a> (<i>seriously, if you haven't read it, do it now, otherwise it's going to be hard</i>). Now we go back in time, to <b>the beginning of the 20th century</b>, deep in the paradox.<br />
<br />
<b>Mathematics</b>, like art, politics, physics, <b>was in a horrible crisis</b>. There was the legitimate doubt that all mathematics was in fact a house of cards. Everything could collapse any minute: people just realized that <b>the math building had no foundation</b>. You know when you are having a deep and meaningful conversation with your friend, but suddenly there is that awkward moment when you realize you were talking about the <a href="https://twitter.com/TheTweetOfGod/status/506313563037007872">Isis</a>, and he was talking about <a href="http://www.slate.com/articles/life/a_fine_whine/2013/02/reclining_airplane_seats_are_a_terrible_idea_and_should_be_banned.html">recliners</a> (you've got to chill out, dude, it's ok to hate them but not <i>that</i> much)? Or when you are well into adulthood, and you realize that all the choices your life is based on where made when you were too much young and immature, and now you are completely disconnected to what once had meaning to you, and it is too late too change, you are trapped in a cage of sorrow build by yourself where you can just float through your meaningless life until OH MY GOD STOP IT THIS IS TOO DEPRESSING.<br />
<br />
Yep, mathematics was in a middle age crisis.<br />
<br />
It works like this:<br />
<br />
What is a triangle? A <b>polygon</b> with three edges and three vertices.<br />
<br />
What is a polygon? A <b>plane</b> figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain or circuit.<br />
<br />
What is a plane? A flat, <b>two-dimensional</b> surface.<br />
<br />
What is a dimension of an object? The minimum number of coordinates needed to specify any <b>point</b> within it.<br />
<br />
What is a point? Ehm... something with no dimensions? No, we defined dimension using points. Then... What?<br />
<br />
You can understand how, in a subject like mathematics that strives to be as objective as possible, <b>the lack of well-defined basic notions was problematic</b>. Before that, they were satisfied with <a href="http://en.wikipedia.org/wiki/Primitive_notion">primitive notions</a>, but with the current level of sophistication it was not enough. <a href="http://en.wikipedia.org/wiki/Hilbert%27s_program">Hilbert</a> tried to solve this with a collective effort: let's write down a list of things we know for sure are true, and such that all the results of mathematics can be proved from those. If something make all the people say "Duh", then it is a right starting point.<br />
<br />
This is how we came up with <a href="http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory">ZFC</a>: a list of nine trivial sentences (like: two sets are the same if they have the same elements. Duh), that form the basis of all ordinary mathematics. Of course there are fringes who don't like it: some people want to drop the C and have ZF, others want ZC or just <a href="http://en.wikipedia.org/wiki/Zermelo_set_theory">Z</a>, others prefer <a href="http://en.wikipedia.org/wiki/KFC">KFC</a>, some guy prefers <a href="http://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory">NBG</a>, or <a href="http://en.wikipedia.org/wiki/Kripke%E2%80%93Platek_set_theory">KP</a>. Oh, and what about <a href="http://en.wikipedia.org/wiki/New_Foundations">New Foundations</a>? It's a mess, but we can say that the majority of people consider ZFC=mathematics.<br />
<br />
But then <a href="http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems">Gödel</a> (25 years old) came and said: careful! Whatever sentences you choose, there will be always mathematical statements that are unprovable, outside the scope of logic! So paradoxical!<br />
<br />
So this is how one can say that something is unprovable in ordinary mathematics: it just cannot be proved from those nine sentences.<br />
<br />
And here come the large cardinals! What if we add to the nine sentences a tenth sentence, like, <b>there exists an inaccessible cardinal</b>. <b>Then we can prove more stuff</b>! Remember: large cardinals are unprovable in ordinary mathematics, so we are adding something genuinely new! But then... something that before was unprovable, now it's provable, using large cardinals! Yay!<br />
<br />
Has it ever happened? It happens every day. The most famous (but controversial) case is <a href="http://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem">Fermat's Last Theorem</a>. When Andrew Wiles proved it, he became a superstar: he was mentioned in Star Trek: Deep Space Nine, in Stieg Larsson's The Girl Who Played with Fire, and a jeans company asked him to pose for an advertisement (he turned it down, unfortunately). Well, the proof uses <a href="http://en.wikipedia.org/wiki/Grothendieck_universe">Grothendieck Universe</a>, and you cannot have a Grothendieck Universe without an inaccessible cardinal. So yes, Wiles used a large cardinal. It is controversial, because many mathematicians claim that it is not necessary to use it, that it is possible to change the proof and avoid it, so maybe Fermat's Last Theorem <i>is</i> provable. (This is granted to start a flame war).<br />
<br />
But there are many other technical results that we know need large cardinals. Also, we have many large cardinals, not just inaccessible, and they are stronger and stronger, therefore lead to more and more mathematical results that are not provable in ordinary mathematics. They have also funny names (I am talking to you, <i>ineffable</i> cardinals), so <b>I'll end with a (partial) <a href="http://en.wikipedia.org/wiki/List_of_large_cardinal_properties">list</a> of them completely without context for your amusement, from the weakest to the strongest</b>:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvW7Wfgo4McR16GPrZuGyMWDwME3mJ5mfVEJxREjL9esH41DNR1dpZOhDiX_LjObw5n45qtHpCECDaurPPqXS9pTexDBSZvkByElVNDfOypG4Db8xHVi_j-d1qtIHko1k1_-eDuaL06A/s1600/Inaccessible+Cardinal.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvW7Wfgo4McR16GPrZuGyMWDwME3mJ5mfVEJxREjL9esH41DNR1dpZOhDiX_LjObw5n45qtHpCECDaurPPqXS9pTexDBSZvkByElVNDfOypG4Db8xHVi_j-d1qtIHko1k1_-eDuaL06A/s1600/Inaccessible+Cardinal.png" height="200" width="320" /></a></div>
<div style="text-align: center;">
Oh, hello, <a href="http://en.wikipedia.org/wiki/Inaccessible_cardinal">inaccessible cardinal</a>!</div>
<div style="text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhb3xS162JlmakwysP5GThBJNnShtiVqy925wdvs4FpVRo6yXLuZYb89plnjz6mWIn8Kg6vyIsdWDnuH8RdHAf5t3W7540pGc-r99ytJL6LGGuLhXLx6nmckgwSwpPNvV0SKeXcWs9IWA/s1600/Reflecting+cardinal.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhb3xS162JlmakwysP5GThBJNnShtiVqy925wdvs4FpVRo6yXLuZYb89plnjz6mWIn8Kg6vyIsdWDnuH8RdHAf5t3W7540pGc-r99ytJL6LGGuLhXLx6nmckgwSwpPNvV0SKeXcWs9IWA/s1600/Reflecting+cardinal.png" height="218" width="320" /></a></div>
<div style="text-align: center;">
<a href="http://en.wikipedia.org/wiki/Reflecting_cardinal">Reflecting cardinal</a></div>
<div style="text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgdIVKRSvt2pHwJqKoXGRBCYmST0_X5gmb_H3oS3zBW6Ebpo1xlKSCc0MkguHsOfSUMYiiNdccaLXaAIL8vxPBrYfobNASwBgc2_bOwr3fAFw3pov30902kjTicCR_D-NaDEJ7R1w6liw/s1600/Measurable+cardinal.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgdIVKRSvt2pHwJqKoXGRBCYmST0_X5gmb_H3oS3zBW6Ebpo1xlKSCc0MkguHsOfSUMYiiNdccaLXaAIL8vxPBrYfobNASwBgc2_bOwr3fAFw3pov30902kjTicCR_D-NaDEJ7R1w6liw/s1600/Measurable+cardinal.png" height="200" width="320" /></a></div>
<div style="text-align: center;">
<a href="http://en.wikipedia.org/wiki/Measurable_cardinal">Measurable cardinal</a></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2ISxeXx3RmsHdKlo4R3_s1RZaOyarKQ1VNp8D9CvWOaMg5wK_vOh58A4KkNso903kX0UP68XmgvyLkhFknXF0Gn5Ea4PbS_mBBmRvz0Lc6IceQ2xf61YBHCZHqXofmkItSWd-nZX5tA/s1600/Zero+dagger.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj2ISxeXx3RmsHdKlo4R3_s1RZaOyarKQ1VNp8D9CvWOaMg5wK_vOh58A4KkNso903kX0UP68XmgvyLkhFknXF0Gn5Ea4PbS_mBBmRvz0Lc6IceQ2xf61YBHCZHqXofmkItSWd-nZX5tA/s1600/Zero+dagger.png" height="200" width="320" /></a></div>
<div style="text-align: center;">
<a href="http://en.wikipedia.org/wiki/Zero_dagger">Zero dagger</a></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj8888rTDbj5mB5UMPIeZ5JCqnyiA-3J-n0MbzsyLOqyeGApM3oE1up8QsW0ajzdpE-pvreKXVs8qGzZ0N-zmAEPysL3rtjBZzuuwkkUx_UDCyihsRrnNHGQ4-AqoCjufGOtdtiyLxvfQ/s1600/Strong+Cardinal.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj8888rTDbj5mB5UMPIeZ5JCqnyiA-3J-n0MbzsyLOqyeGApM3oE1up8QsW0ajzdpE-pvreKXVs8qGzZ0N-zmAEPysL3rtjBZzuuwkkUx_UDCyihsRrnNHGQ4-AqoCjufGOtdtiyLxvfQ/s1600/Strong+Cardinal.png" height="200" width="320" /></a></div>
<div style="text-align: center;">
<a href="http://en.wikipedia.org/wiki/Strong_cardinal"> Strong cardinal</a></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgztvQ7toxNfYBn2we_S80bINedHBAD7iZg7KSXR_BAEZ-XFHCtPdBuUbOJkh11YXMTw45eeM89BUT_3di8u1QtoB1THRBM_UPf41f1GgPqZDIJRDQ814fg3wvjVz1_cjzczie9hSpfFg/s1600/Zero+pistol.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgztvQ7toxNfYBn2we_S80bINedHBAD7iZg7KSXR_BAEZ-XFHCtPdBuUbOJkh11YXMTw45eeM89BUT_3di8u1QtoB1THRBM_UPf41f1GgPqZDIJRDQ814fg3wvjVz1_cjzczie9hSpfFg/s1600/Zero+pistol.png" height="200" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
Zero pistol</div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div style="text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgERSW2uoW0CR26_gBVjxwbvmEJOC5RBh6LF8iNAeMK457tyBCoQxpygqRSivqd7Xg5TFuiGDq3pK3liHXwy583DfNqaqHxWWSm64_o9EsvjMnOoa1fuoQC9T9JWj-DZzep_BvicTzkAA/s1600/Supercompact+cardinal.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgERSW2uoW0CR26_gBVjxwbvmEJOC5RBh6LF8iNAeMK457tyBCoQxpygqRSivqd7Xg5TFuiGDq3pK3liHXwy583DfNqaqHxWWSm64_o9EsvjMnOoa1fuoQC9T9JWj-DZzep_BvicTzkAA/s1600/Supercompact+cardinal.png" height="200" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://en.wikipedia.org/wiki/Subcompact_cardinal">Supercompact cardinal</a></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgp-LvCQcopTFzIVb59EsZjcEGH12GdgX_Ct5vrDsxxCfdsh0f9vdrjeFK6Ew-VzaMLKbyJvegFQKBK4tAFbPsxY9OGU8c7S0_tKtLmKK4NjvP-n7P83L7Pit8_tiFylAfT3BTln5ADZw/s1600/Huge+cardinal.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgp-LvCQcopTFzIVb59EsZjcEGH12GdgX_Ct5vrDsxxCfdsh0f9vdrjeFK6Ew-VzaMLKbyJvegFQKBK4tAFbPsxY9OGU8c7S0_tKtLmKK4NjvP-n7P83L7Pit8_tiFylAfT3BTln5ADZw/s1600/Huge+cardinal.png" height="200" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://en.wikipedia.org/wiki/Huge_cardinal">Huge cardinal</a></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEirw-rrBiyhUikMnQPdCGPIsbn9zmRoq-DqWjOjN7qqpWBYYWAa-zmdH79xdX9qCtLv15wy2yX3G5EV5DAkYb96tiduLjHxfIkaPGd0IP09GGi__5gzP9maolwh7BsDSQapmZGfr_sitg/s1600/I0.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEirw-rrBiyhUikMnQPdCGPIsbn9zmRoq-DqWjOjN7qqpWBYYWAa-zmdH79xdX9qCtLv15wy2yX3G5EV5DAkYb96tiduLjHxfIkaPGd0IP09GGi__5gzP9maolwh7BsDSQapmZGfr_sitg/s1600/I0.png" height="200" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://en.wikipedia.org/wiki/Rank-into-rank">I, 0</a></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjRi1dGiR7IhDLY8wxalLsX5s8DAPtOIv_1czCxM9cDYJNaRCEK-XKQBGNCEQUTg7oSRYZlZP3e4oreDxuvf6HESmbqebcUguV3xDDSNyzXDlcAnZQ_YmMmjizP21V2rZYUlFiFfldSDw/s1600/Icarus+sets.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjRi1dGiR7IhDLY8wxalLsX5s8DAPtOIv_1czCxM9cDYJNaRCEK-XKQBGNCEQUTg7oSRYZlZP3e4oreDxuvf6HESmbqebcUguV3xDDSNyzXDlcAnZQ_YmMmjizP21V2rZYUlFiFfldSDw/s1600/Icarus+sets.png" height="200" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
Icarus sets</div>
<br />
<br />
So, good job, Artemis! If the problem you're thinking of is unprovable, then using large cardinals is the most sensible thing you can do. Start with an inaccessible cardinal, and if it is not enough, go up up up, until you find something useful.<br />
<br />
And now you can try this at home too! (*) Enjoy your time solving unprovable theorems with large cardinals!<br />
<br />
<span style="font-size: x-small;">(*) Keep out of reach of children.
Keep away from food, drink and animal foodstuffs. <br />When using do not smoke. Avoid contact with eyes.
If swallowed, seek medical advice immediately and show this container or label. </span><br />
<div class="separator" style="clear: both; text-align: left;">
<br /></div>
<div class="separator" style="clear: both; text-align: left;">
<i>You want to know what is the use of large cardinals? How about... <a href="http://cantorontheshore.blogspot.it/2014/09/god-and-big-bang-theory.html">proving God</a>? Or just waisting a lot of time for an <a href="http://cantorontheshore.blogspot.it/2014/10/one-reinhardt-and-counting.html">addicting game</a>, that is important too.</i></div>
Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com0tag:blogger.com,1999:blog-8155036766532593680.post-55217354794841959462014-09-08T12:39:00.001-07:002015-02-24T08:55:28.018-08:00Artemis Fowl and the large cardinals paradox (Part one)Let's go back to young adults literature.<br />
<br />
This time I am picking a sentence from the book <a href="http://en.wikipedia.org/wiki/Artemis_Fowl:_The_Time_Paradox">Artemis Fowl: the Time Paradox</a>.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRq6S89IfupK1DA_Af7Zk2fKN5qLQmBSWyKMq2Ekx16hX3Zf59Vcw91O7bn0KG_zYJ9CC2n6Wwk3hkCkAkvcjAMqUjF1XvNlZqqlMosvIRKgfPDdLr7x0ZBp-C1h4QzSnXa7NlSkkCWA/s1600/ArtemisFowlTimeParadoxCover.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRq6S89IfupK1DA_Af7Zk2fKN5qLQmBSWyKMq2Ekx16hX3Zf59Vcw91O7bn0KG_zYJ9CC2n6Wwk3hkCkAkvcjAMqUjF1XvNlZqqlMosvIRKgfPDdLr7x0ZBp-C1h4QzSnXa7NlSkkCWA/s1600/ArtemisFowlTimeParadoxCover.gif" height="320" width="210" /></a></div>
<br />
It's the sixth book in the series <a href="http://en.wikipedia.org/wiki/Artemis_Fowl_(series)">Artemis Fowl</a>, by the talented <a href="http://en.wikipedia.org/wiki/Eoin_Colfer">Eoin Colfer</a>. I won't compare it with Harry Potter, like everybody does, first because he dislikes it (it is like comparing an apple with an orange, <a href="http://www.theguardian.com/books/2001/may/13/booksforchildrenandteenagers.features">he said</a>), second because it is much more unique: it is a mixture of sci-fi technology and typical fantasy lore, and the main character is not a whiny do-gooder but an unpleasant sociopathic teenager, extremely intelligent and sort-of-criminal. I feel him. The books are full of action, but they reward the brainy young reader.<br />
<br />
So, at a certain point Artemis Fowl in this book is time-traveling (I hope it's not a spoiler, but come on, it is in the title!), and during his travel he's looking for ways to fill this time. How does a genius spend his time? Like this:<br />
<br />
<span style="font-family: Courier New, Courier, monospace;">He focused on the high end of his intellect, solving unprovable theorems with large cardinals and composing an ending for Schubert's unfinished Symphony N° 8.</span><br />
<span style="font-family: Courier New, Courier, monospace;"><br /></span>
<span style="font-family: inherit;">Yes! YES! Infinitytimesinfinity times yes! This is so exact it hurts. <i>Oh boy</i>. I am not talking about the decades-long row on whether the <a href="http://en.wikipedia.org/wiki/Symphony_No._8_(Schubert)">Unfinished Symphony</a> is the 8th or the 7h, of course (for this, go to schubertontheshore.blogspot.com), but I am talking about large cardinals and unprovability! In fact, I think that in the whole world there are just, like, two thousands people able to come up with that sentence? Who tipped Eoin Colfer? I want to know!</span><br />
<span style="font-family: inherit;"><br /></span>
Well, be prepared, now, after reading this you will be one of those!<br />
<br />
Let me start with large cardinals. What are these?<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXgwIaOW6CbTNvbfLBbBTQPs1g4YbiBX8YcILzWIQgiY734D4yk-Nr3s2CZriZ8M9_2Ujhe1ptIDOrvGLjyM8HL-CKuJ8F-z7W81uPzBRV0eZcySysKbjCVfPNg9pqFTO7nmJIr22K8w/s1600/Large+Cardinal.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXgwIaOW6CbTNvbfLBbBTQPs1g4YbiBX8YcILzWIQgiY734D4yk-Nr3s2CZriZ8M9_2Ujhe1ptIDOrvGLjyM8HL-CKuJ8F-z7W81uPzBRV0eZcySysKbjCVfPNg9pqFTO7nmJIr22K8w/s1600/Large+Cardinal.png" height="200" width="320" /></a></div>
<br />
<div style="text-align: center;">
<span style="font-size: x-small;">Pictured: not what I am talking about</span></div>
<br />
<b>Cardinals are just math-speech for numbers and infinities.</b> In practice, <b>they are the measure of the largeness of a set</b>: if a set is finite it will be a number (finite cardinals), if it is infinite it will be some infinity (infinite cardinal).<br />
<br />
Wait, if this is the first time you read this blog, then maybe you are not aware of Cantor's Theorem. Read <a href="http://en.wikipedia.org/wiki/Cantor%27s_theorem">this</a>, and be prepared to have your mind blown. Or just don't read it, and believe me: there are many infinities, some bigger, some smaller, and not just one. Booom.<br />
<br />
But then what can be considered <i>large</i> for the people that are used to handle infinities like it's nothing? <a href="http://cantorontheshore.blogspot.co.at/2014/08/advanced-thinking-thinking-infinity.html">Here</a> you can find some infinite sets that needs some serious brainpower to be visualized, and yet they are all <i>extremely</i> small. We have to pick an <b><a href="http://en.wikipedia.org/wiki/Inaccessible_cardinal">inaccessible cardinal</a>, the smallest of the large cardinals</b>, to just start to realize how large these guys are.<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBh2yC4KAdEVURcMBluVzH60xwe5HfFLsm_3gRwIxB8IigU-LPb1UewM49cSx97M80YqF3KseYEynLaw68CpGJOjgsAR6lx5dn1KPdMF0FJrfKiNO9Dq8A06aWOiQuBOS_xfVZ_-JLog/s1600/Inaccessible+Cardinal.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBh2yC4KAdEVURcMBluVzH60xwe5HfFLsm_3gRwIxB8IigU-LPb1UewM49cSx97M80YqF3KseYEynLaw68CpGJOjgsAR6lx5dn1KPdMF0FJrfKiNO9Dq8A06aWOiQuBOS_xfVZ_-JLog/s1600/Inaccessible+Cardinal.png" height="200" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div style="text-align: center;">
<span style="font-size: x-small;">Again, not what I am talking about. Stop it.</span></div>
<br />
Think: how many infinities there are under the first infinite cardinal? 0, that was a trick question. How many under the second infinite cardinal? 1. Well, under an inaccessible cardinal, there are <i>as many as the cardinal itself</i>. Woah, that's big. Really big. In a certain sense, it is an infinity so powerful that all the smaller ones are like finite (*). It's really a jump in another dimension.<br />
<br />
<b>It is so big, that it trascends mathematics.</b> With that I mean that it is completely outside mathematics, it cannot be reached with proofs. <b>One cannot prove that it exists.</b> And here we are with unprovability, the other key word in Artemis Fowl's thought. Please remark: he says unsolvable, not unsolved. He is not trying to prove theorems no one else ever proved, <b>he is trying to prove theorems it is <i>impossible</i> to prove.</b> How come?<br />
<br />
To understand this, we must do like Artemis Fowl and go back in time, precisely in the beginning of the 20th century! Prepare all the luggage, next Monday we start!<br />
<br />
Cliffhanger!<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMfIfjfxP7l1v62AUisnUqjE3DbSBvQ127zWi20NLHU7fxfKi1Vf6vM11YaFmAchGS5HJeRB-z43m3npPZSMCKXns8xakjhC_Wl-WKzIsUV3FwlVG4B0gg9j6eBCQ2ICIiPnMOm0hidw/s1600/Cliffhanger.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMfIfjfxP7l1v62AUisnUqjE3DbSBvQ127zWi20NLHU7fxfKi1Vf6vM11YaFmAchGS5HJeRB-z43m3npPZSMCKXns8xakjhC_Wl-WKzIsUV3FwlVG4B0gg9j6eBCQ2ICIiPnMOm0hidw/s1600/Cliffhanger.png" height="200" width="320" /></a></div>
<br />
<br />
(*) I am a mathematician, so I cannot leave this sentence so vague. This is what I mean: if you have around you just a finite amount of finite objects, you cannot reach infinity. No way. You can add them, multiply them, stack them, exponentiate them, the result will always be finite. The same thing with an inaccessible cardinal: if you have less than an inaccessible cardinal objects around you, you cannot "reach" the inaccessible. Yep, this is why it's called inaccessible. Duh.<br />
<br />
<i>Aaaand, here is the <a href="http://cantorontheshore.blogspot.it/2014/09/artemis-fowl-and-large-cardinals_22.html">second part</a>! </i>Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com0tag:blogger.com,1999:blog-8155036766532593680.post-43597289071328100822014-08-25T13:40:00.000-07:002015-02-24T08:49:21.298-08:00Infinity times infinity: updatesRemember <a href="http://cantorontheshore.blogspot.co.at/2014/08/advanced-thinking-thinking-infinity.html">this</a> post? The one about infinity times infinity? The kindergarten kids? The infinities of dots? The pickets? That one.<br />
<br />
Well, I have updates on that.<br />
<br />
UPDATE 1:<br />
<br />
Just as I published the post, I received a notification from Google+. Weird, Google+ is a wasteland (sorry Google, I know you're doing your best). What is this scream from the void?<br />
<br />
Google+ automatically uploaded this gif for me:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCNQGL1nU42vmRcaOcYfqrWK42BM7EBkeERxGRew-ghAUZen5USO6JSN4xwnbr-S7UrOQtsbmo07fDnHsNMERkdx-VtAAiNjctubBgN3gT6bTiq2n2NXk14sxy7vNwJc9kkByvTU6_MA/s1600/Infinity-MOTION.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCNQGL1nU42vmRcaOcYfqrWK42BM7EBkeERxGRew-ghAUZen5USO6JSN4xwnbr-S7UrOQtsbmo07fDnHsNMERkdx-VtAAiNjctubBgN3gT6bTiq2n2NXk14sxy7vNwJc9kkByvTU6_MA/s1600/Infinity-MOTION.gif" height="50" width="320" /></a></div>
<br />
Very cool gif, Google+! Thank you! This explains my thoughts much, much better. I don't know how you came up with this.<br />
<br />
Wait.<br />
<br />
<b>How <i>did</i> you come up with this?</b> Not only this animation makes much sense, but it is also named infinity-MOTION. Perfect name. Is there a brilliant man/woman at Google+'s that follows <i>all</i> my posts, looks up at the pictures and creates better ones? To make me feel inferior? Does he/she follows me around, stalking every keystroke?? You're not better than me, <i>person</i>!<br />
<br />
Or maybe I am overreacting. It is just an automatic software. That crawls through all my stuff. And makes creepily appropriate gifs. Is it... is this how Skynet started? With <i>gifs</i>? <i>What are you doing to my life, Google+</i>?!<br />
<br />
UPDATE 2:<br />
<br />
It turns out that <a href="https://www.google.com/search?q=infinity+times+infinity&rlz=1C1TEUA_enAT506AT506&espv=2&source=lnms&tbm=isch&sa=X&ei=Ny_6U4nmEaSW0QXd4oCgDQ&ved=0CAYQ_AUoAQ&biw=1455&bih=705">infinity times infinity</a> is a thing, and there is a symbol for it.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbbOvRP3wiXR0X0prMQZwoJKcAsn0-DZRKQqKn4FX5DzA5i5cKDRg7-qrrAmlSMXdEznXNpABwyzDbcAxAo7GlOp2FLHoUSEeUiW6E3T9OMcsdMH1xc9fPpCxeSNWbFUvSjutjHHk04g/s1600/33284484716147693_99uiTSqw_c.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbbOvRP3wiXR0X0prMQZwoJKcAsn0-DZRKQqKn4FX5DzA5i5cKDRg7-qrrAmlSMXdEznXNpABwyzDbcAxAo7GlOp2FLHoUSEeUiW6E3T9OMcsdMH1xc9fPpCxeSNWbFUvSjutjHHk04g/s1600/33284484716147693_99uiTSqw_c.jpg" height="320" width="320" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<br /></div>
<div class="separator" style="clear: both; text-align: justify;">
It seems a must for romantic things (as in "I love you infinity times infinity") and it appears in jewellery, tattoos, cards, etc... First, it seems like this blog will never be free from romantic stuff. This is great, as a good part of my intended audience is clearly made of teenage girls and pickup artists. Second, I am not really sure about this symbol. I mean, it is clearly one infinity (<a href="http://en.wikipedia.org/wiki/Lemniscate">lemniscate</a>) on top of the other, so <b>it is more like infinity <i>plus</i> infinity.</b> I imagine infinity times infinity as something more complex, a bit more mindblowing, something like this?</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTPto2poWEh2NGrIF_rg1Vr4nHl-2wldQjTkMm2LxqR-An8-a5HbkOrnjiEpF5WH4IoR-qiRkZKEiAUtdIN8PEPxwV6KZMAnWVetr102bQANFazso35uvdK8OyT9SlZd9nwfrN5S07Bw/s1600/Infinitytimesinfinitysymbol.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTPto2poWEh2NGrIF_rg1Vr4nHl-2wldQjTkMm2LxqR-An8-a5HbkOrnjiEpF5WH4IoR-qiRkZKEiAUtdIN8PEPxwV6KZMAnWVetr102bQANFazso35uvdK8OyT9SlZd9nwfrN5S07Bw/s1600/Infinitytimesinfinitysymbol.png" height="320" width="282" /></a></div>
<div class="separator" style="clear: both; text-align: justify;">
</div>
Well, this seems too much a pair of angry eyes. Maybe something more fractal? I am not a graphic designer (as you can see very well from my pictures), so please, <b>somebody invent a better symbol</b>! Infinity times infinity deserves it!<br />
<br />
UPDATE 3<br />
<br />
This.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/xbO61Nd0Jlk?feature=player_embedded' frameborder='0'></iframe></div>
<br />
I have nothing to say about this. Everything he said it's right.<br />
<br />
<i>Did anybody said <a href="http://cantorontheshore.blogspot.com/2014/12/and-best-tattoo-of-year-2014-is.html">tattoo</a>? Also, if you spend time on Google+, probably you have a lot of time for <a href="http://cantorontheshore.blogspot.com/2014/10/one-reinhardt-and-counting.html">this</a>.</i>Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com2tag:blogger.com,1999:blog-8155036766532593680.post-61203955654239719522014-08-18T00:15:00.000-07:002015-02-24T08:44:27.921-08:00These are infinities that count!It's summer! Time to relax, chill out, go to the beach (at least this is what I heard normal people do, while I am in my without-air-conditioning city office explaining to various research centers the impact of my research on European society), listen to summer hits:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/Mha6vCSrWQE?feature=player_embedded' frameborder='0'></iframe></div>
<br />
This is "Countably Infinite" by the <a href="http://www.theagtrio.com/">A.G.Trio</a>, an electrohouse band from Austria. In summer 2012 (I said summer hits, I didn't specified the year) it peaked the Austrian chart, and it also entered the German Club Chart. But I am not interested in the music now, I am interested in the lyrics. As far as I have understood, they're about a couple breaking up (of course, like <i>every</i> song ever written, so nothing to see here), the key sentence that gives the title of the song is this:<br />
<br />
<span style="font-family: Courier New, Courier, monospace;">There are countably infinite things I'd like to say to you</span><br />
<br />
In this cases I think: <b>what does a person that doesn't know what "countably infinte" means feels hearing this?</b> Which images come to the mind? An infinite... you can count? Because in fact <b>"countably infinite" is a pretty specific term in set theory</b>.<br />
<br />
<b>Something is countably infinite if it is infinite and as big as the natural numbers</b>, or <a href="http://cantorontheshore.blogspot.co.at/2014/07/a-pint-of-aleph.html">aleph 0</a>. That is, if you can connect every object to one and only one number, the objects are countably infinite. In this way, you can <i>count </i>the objects: object 1, object 2, object 3, ... and this is way is called "countably". <b>One way to do this is a list</b>: if you can write the objects, one every line, without ever stopping, than you have countably infinite.<br />
<br />
So, <b>is it possible for the singer to have countably infinite things to say?</b> Let's try to imagine the possible things he has to say as a list:<br />
<br />
1. I love you<br />
3. Don't leave<br />
2. I'll miss you<br />
3. My, that is a BIG pimple<br />
4. It's like... purple<br />
5. Hey, did you know that there is no word that rhymes with purple?<br />
6. The sky is blue<br />
7. I kissed a girl, and I liked iiit (of course)<br />
8. Some infinities are bigger than other infinities...<br />
9. BAM! <i>Romantic</i><br />
10. There are more stars in the universe than grains of sand on all the beaches of Earth<br />
11. I love <a href="http://www.amazon.com/Taken-T-Rex-Dinosaur-Erotica-Christie-ebook/dp/B00FI9JFFO">dinosaur erotica</a><br />
12. No, no, no this is not related to the first sentence, don't leave!<br />
13. Where am I?<br />
14. <a href="http://en.wikipedia.org/wiki/I_Am_Sitting_in_a_Room">I am sitting in a room</a>, different from the one you are in now<br />
15. But seriously, how would you say your skin is <i>scaly</i> from 0 to 10?<br />
16. ...<br />
<br />
Wait, this is taking too long. Let's do this more systematically. How many thoughts there are, at all? Better: <b>let's count how many sentences one can write</b>. For example: how many with one character? we have the letters (26) and the space (1). Who cares about punctuation. So 27. How many with two characters? 27 times 27, that is 729. And so on. So let's code any sentence with a number: a=1, b=2, z=26, _=27, aa=28, ab=29, a_=54, ba=55 and so on. Think about it: all the words you are seeing are saved in the computer as binary numbers, so of course to <b>every sentence is associated one and one only number, and to every number one and one only sentence</b>. For example: the number 1234567890 is the sentence "ceaantr" (almost Cantor!), while "I_love_you" is the number 76387629278892. So there <i>are</i> countably infinite possible things to say, one for each number!<br />
<br />
But does the singer have really infinite things to say? Our brain is pretty big, but not infinite, it has somewhat around <a href="http://www.scientificamerican.com/article/what-is-the-memory-capacity/">2.5 petabytes</a> (or a million gigabytes). So<b> the thoughts he has in mind have to be finite</b>! Come on, A.G. Trio! How could you even think that one guy had infinite thoughts! It's like you are purposedly exaggerating for...<br />
<br />
Wait a minute.<br />
<br />
That's it, isn't it? They didn't <i>really</i> mean infinite things. It was just a way to say "more things that I can say", wasn't it? And I am making a fool of myself meticulously analyzing a simple hyperbole?<br />
<br />
...<br />
<br />
Blimey! I always fall for this! I... I need time to think.<br />
<br />
(Anyway, Cantor is 43868727)<br />
<br />
<i>Countable infinity is probaby much, much <a href="http://cantorontheshore.blogspot.com/2014/08/advanced-thinking-thinking-infinity.html">bigger</a> than what you have in mind. Still, other things are <a href="http://cantorontheshore.blogspot.com/2015/01/infinite-mess-part-two.html">even bigger</a>.</i>Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com0tag:blogger.com,1999:blog-8155036766532593680.post-60622073415979364672014-08-11T00:20:00.000-07:002015-02-24T08:38:15.693-08:00Advanced Thinking & Thinking: Infinity times Infinity<i>Warning: the procedures in this post, if followed correctly, can lead to vertigo, psychological intoxication, and air-headedness. In other words: awesomeness.</i><br />
<br />
I wanted to give you a tiny, little glimpse of the sensations that I experience daily on working in Set Theory. I want to take you on a stroll in the <a href="http://en.wikiquote.org/wiki/David_Hilbert" target="_blank">paradise</a> that Cantor provided us. <b>I am going to describe some really simple infinite set, and the only thing you have to do is to try to imagine it.</b> Sometimes the mental image just don't come: don't worry, just keep thinking about it, and return back to it in another moment. Also, take your time. Don't just say "Yes, I understood" and go on, try to explore with your mind every nook and cranny of the infinite set.<br />
<br />
And have the right attitude. Which attitude, you say? This one:<br />
<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/Nfc7G4xKsl8?feature=player_embedded' frameborder='0'></iframe></div>
<br />
This advertisement is incredible.<b> It manages to introduce effortlessy a nice amount of infinite sets. But can you really image infinity times infinity?</b> Try it now. I'll wait.<br />
<br />
...<br />
<br />
What took you so much? So let's see if my description is the same as yours.<br />
<br />
This is one.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg37xKHjeUIRP92-uyPAO7IOCnk3asIM28R6GN92TlaFuJEH6AyObNCC4Xxk1f60VqDwOrgHXbLtyxFZSBuqPn7qmJfTpSJXhHsbRIJ9n3stGZ1C-TV46ch2oI_h_xG1DziolNnNzGR4Q/s1600/One.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg37xKHjeUIRP92-uyPAO7IOCnk3asIM28R6GN92TlaFuJEH6AyObNCC4Xxk1f60VqDwOrgHXbLtyxFZSBuqPn7qmJfTpSJXhHsbRIJ9n3stGZ1C-TV46ch2oI_h_xG1DziolNnNzGR4Q/s1600/One.jpg" height="50" width="320" /></a></div>
<br />
<br />
This is two (yeah, I know, stay with me).<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKaIxx29bV2YJJeXQw-ROpnYQuQe9ZgvypQcDsa67vZrqkkdGjmly36tvoIHZocafCdmWUwInBU69dS-74YdD2doz8Sssh_w44pUyp7OOD7KikQrY3ZopRxNd9qEv3FHuGgz-2rJ-VJQ/s1600/Two.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKaIxx29bV2YJJeXQw-ROpnYQuQe9ZgvypQcDsa67vZrqkkdGjmly36tvoIHZocafCdmWUwInBU69dS-74YdD2doz8Sssh_w44pUyp7OOD7KikQrY3ZopRxNd9qEv3FHuGgz-2rJ-VJQ/s1600/Two.jpg" height="50" width="320" /></a></div>
<br />
<br />
And so on. Just adding one on the right, you are making +1.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIXNdCaZjZUXdGwFrdaka0Xdsn519KuZmFuNL9oHNg7bxOELwmzlow2x7XyjE506Xpgg38oLrRB0e0lqK9qTi8v8DqCrWjKNBuEDlfD7yRnfaowEL2IlSLjSL5zmP0nS0oJqtHW6kKZw/s1600/Three.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIXNdCaZjZUXdGwFrdaka0Xdsn519KuZmFuNL9oHNg7bxOELwmzlow2x7XyjE506Xpgg38oLrRB0e0lqK9qTi8v8DqCrWjKNBuEDlfD7yRnfaowEL2IlSLjSL5zmP0nS0oJqtHW6kKZw/s1600/Three.jpg" height="50" width="320" /></a></div>
<br />
Once you finished the numbers, you have the first infinite set! Congratulations! We call it ω. (<b>Infinity</b>! like the girl said).<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7jWhsStyEcq2LTB_-WCMqNTTvWhaS4tE2IJARG7y27n_BDS1tr9__Lp0bjlIEYORI078awPLqoXRb2Gnmz4Tg9dkTLIARbHAKt_eqzk8Uv6LAQHzN3NAJBZKbzQWGsH07JJmHK57swA/s1600/Infinity.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7jWhsStyEcq2LTB_-WCMqNTTvWhaS4tE2IJARG7y27n_BDS1tr9__Lp0bjlIEYORI078awPLqoXRb2Gnmz4Tg9dkTLIARbHAKt_eqzk8Uv6LAQHzN3NAJBZKbzQWGsH07JJmHK57swA/s1600/Infinity.jpg" height="50" width="320" /></a></div>
<br />
Now, you know how to do ω+1: after ω, add one point.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZB5wb8M-RHcYI75o4Aq0inZ9muExkRVFeozJwOPlxlsUwRXA31ar6nxQdljcXErYS5p5s6te8zoikI93MXO7Je7stmcxWG712Dn_dmVtxulZ14uK_a3-ozISLjFUgszLlguvyhv0wfg/s1600/Infinity+plus+one.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZB5wb8M-RHcYI75o4Aq0inZ9muExkRVFeozJwOPlxlsUwRXA31ar6nxQdljcXErYS5p5s6te8zoikI93MXO7Je7stmcxWG712Dn_dmVtxulZ14uK_a3-ozISLjFUgszLlguvyhv0wfg/s1600/Infinity+plus+one.jpg" height="50" width="320" /></a></div>
<br />
Well, there is not much space there, let's go to a new line.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrU72VUfimNg4FZxXS2-oe4patRmUxHDWgc-5LJvMlymmGQQuFMZ7K6tsSljIp8AHDSlzFPQip80Su8rUCbm_V8WIb7KTgKcLpS1EjDdGtu0agCoA-96lHJK4uJUmoUtY2Ewk8myu7Dg/s1600/Infinity+plus+one+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrU72VUfimNg4FZxXS2-oe4patRmUxHDWgc-5LJvMlymmGQQuFMZ7K6tsSljIp8AHDSlzFPQip80Su8rUCbm_V8WIb7KTgKcLpS1EjDdGtu0agCoA-96lHJK4uJUmoUtY2Ewk8myu7Dg/s1600/Infinity+plus+one+2.jpg" height="75" width="320" /></a></div>
<br />
And now add another one (ω+2).<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiK3mfWhZYX08oyAdogbU0UzI9znnozqddw5pR6O-29jAk4crwDJoeYfUWQOssBPGCyqpSLwq_PGp4NB7z5i5whbYEB4pehier4T7arRSOlCfwuKBdGS41rDkhq3LX-y1oeuB9kMChwug/s1600/Infinity+plus+two.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiK3mfWhZYX08oyAdogbU0UzI9znnozqddw5pR6O-29jAk4crwDJoeYfUWQOssBPGCyqpSLwq_PGp4NB7z5i5whbYEB4pehier4T7arRSOlCfwuKBdGS41rDkhq3LX-y1oeuB9kMChwug/s1600/Infinity+plus+two.jpg" height="75" width="320" /></a></div>
<br />
And so on, until you finished the numbers again. We have ω+ω! (<b>Infinity plus infinity</b>, like the guy said).<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlpJTiPBQiSP2GdLDAFtjL5M8mbhJ9nCwjh0QW3RGgWeDUtt0OoEOzct2TyQfrGES1OnGkf3LwhaLHOBJE4hGgvjEJnzalskMyXXZ3OZ-n_-_BHg9bSfUQq6GOR3pSwTS8x0e2d6ZPyg/s1600/Infinity+plus+infinity.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlpJTiPBQiSP2GdLDAFtjL5M8mbhJ9nCwjh0QW3RGgWeDUtt0OoEOzct2TyQfrGES1OnGkf3LwhaLHOBJE4hGgvjEJnzalskMyXXZ3OZ-n_-_BHg9bSfUQq6GOR3pSwTS8x0e2d6ZPyg/s1600/Infinity+plus+infinity.jpg" height="75" width="320" /></a></div>
<br />
Now add another point, and continue. We have ω+ω+ω, i.e. three times infinity.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj8R4B5unoV0tf2zj_m4CxXlldzkJU_L2-FcTJySGhdGdu2777pBciOags17dl0DC9uyVhld5clrQ5iUK_xrPwY8tjjAqktSSlJDTrapYIB1ust-4MqYxX167Isa1hrIQ1SQwXtUvoZ-g/s1600/Infinity+plus+infinity+plus+infinity.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj8R4B5unoV0tf2zj_m4CxXlldzkJU_L2-FcTJySGhdGdu2777pBciOags17dl0DC9uyVhld5clrQ5iUK_xrPwY8tjjAqktSSlJDTrapYIB1ust-4MqYxX167Isa1hrIQ1SQwXtUvoZ-g/s1600/Infinity+plus+infinity+plus+infinity.jpg" height="75" width="320" /></a></div>
<br />
Go on, with four times infinity, five times infinity, and so on. At the end we have ωxω (<b>Infinity times infinity</b>! like the other girl said).<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhp0xjxafR_g6tlRVBoaqamNwyeLUin0ZHTHR-wvUHJ2SU8C5AIQmBDZude58dQiztRfgmKf_SKqobkaJPZyybsfLKoh26rooA9IduYTZhkI1VARIMaxZjRWY5EadqSqT8JIPr9EHmtrA/s1600/Infinity+times+infinity.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhp0xjxafR_g6tlRVBoaqamNwyeLUin0ZHTHR-wvUHJ2SU8C5AIQmBDZude58dQiztRfgmKf_SKqobkaJPZyybsfLKoh26rooA9IduYTZhkI1VARIMaxZjRWY5EadqSqT8JIPr9EHmtrA/s1600/Infinity+times+infinity.jpg" height="320" width="320" /></a></div>
<br />
<br />
Great, this was not <i>that</i> difficult. So let's go deeper. I want you to imagine <span style="font-family: Calibri, sans-serif; font-size: 11pt; line-height: 115%;">ω</span><sup style="font-family: Calibri, sans-serif; line-height: 115%;">ω</sup>, i.e. <b>Infinity to the power of infinity</b>! Let's do it visually.<br />
<br />
Visualize again <span style="font-family: Calibri, sans-serif; font-size: 11pt; line-height: 115%;">ω</span>.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGu60IYyBqtespe3eC-RzyP_9tP65miTp6NyyAzTeR-LT780zyTeLvIMkFcoXpIu_hqPziFQf3YYQ6FUnGBIgKqnIv3FnW1q5bon2qP2Bwfa96xCpuRuyZC1O_BTQRkLMfMbpulKDyxg/s1600/Infinity+(again).jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGu60IYyBqtespe3eC-RzyP_9tP65miTp6NyyAzTeR-LT780zyTeLvIMkFcoXpIu_hqPziFQf3YYQ6FUnGBIgKqnIv3FnW1q5bon2qP2Bwfa96xCpuRuyZC1O_BTQRkLMfMbpulKDyxg/s1600/Infinity+(again).jpg" height="50" width="320" /></a></div>
<br />
Now, between each two points, add <span style="font-family: Calibri, sans-serif; font-size: 11pt; line-height: 115%;">ω</span>. We have ωxω, i.e. <span style="font-family: Calibri, sans-serif; font-size: 11pt; line-height: 115%;">ω</span><sup style="font-family: Calibri, sans-serif; line-height: 115%;">2</sup>.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3XeqqDNBAjk7afZsm3qwevBatLxba41O6PtolPld6zMWgsgyNjatLsC01O8_j1DoEzUrBe9h9SIf5GyQkOCNzZjn0LPbNGNcvoh-6sQ3AjIlGczg0Dq82_aie7paoKeR7kLKayMz4gA/s1600/Infinity+times+infinity+(again).jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3XeqqDNBAjk7afZsm3qwevBatLxba41O6PtolPld6zMWgsgyNjatLsC01O8_j1DoEzUrBe9h9SIf5GyQkOCNzZjn0LPbNGNcvoh-6sQ3AjIlGczg0Dq82_aie7paoKeR7kLKayMz4gA/s1600/Infinity+times+infinity+(again).jpg" height="50" width="320" /></a></div>
<br />
Do it again: between any two points in <span style="font-family: Calibri, sans-serif; font-size: 11pt; line-height: 115%;">ω</span><sup style="font-family: Calibri, sans-serif; line-height: 115%;">2</sup>, add ω. We have <span style="font-family: Calibri, sans-serif; font-size: 11pt; line-height: 115%;">ω</span><sup style="font-family: Calibri, sans-serif; line-height: 115%;">3</sup>.<br />
<br />
Of course I am not going to make a drawing for that, don't be <i>silly</i>. This is the moment when we have to use just our minds, the screen does not have enough resolution for that. Now do it again (add one <span style="font-family: Calibri, sans-serif; font-size: 11pt; line-height: 115%;">ω</span> for each point in <span style="font-family: Calibri, sans-serif; font-size: 11pt; line-height: 115%;">ω</span><sup style="font-family: Calibri, sans-serif; line-height: 115%;">3</sup>), and again (<span style="font-family: Calibri, sans-serif; font-size: 11pt; line-height: 115%;">ω</span><sup style="font-family: Calibri, sans-serif; line-height: 115%;">4</sup>), and again! In the end you have in front of you <span style="font-family: Calibri, sans-serif; font-size: 11pt; line-height: 16.866666793823242px;">ω</span><sup style="font-family: Calibri, sans-serif; line-height: 15.333333015441895px;">ω</sup>!<br />
<br />
Let's try another way. Imagine you are on a road, with pickets on the side of the road. The first picket you see is one meter high. You start counting them: one, two, three...<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUGzVUr-ptmj6uOZiGcNSDC9Oi3dPHxPwnAUC2CB3U_UM7-RiMKLnOoovJ_SoMjZth2WSZLy0ZeHfUBlYRn4PqbY5_7KzFrbYChksFJPxfdWuYQCa6Vh5omKCND0NBpe_tyCUFFX5ieQ/s1600/Road+to+infinity.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUGzVUr-ptmj6uOZiGcNSDC9Oi3dPHxPwnAUC2CB3U_UM7-RiMKLnOoovJ_SoMjZth2WSZLy0ZeHfUBlYRn4PqbY5_7KzFrbYChksFJPxfdWuYQCa6Vh5omKCND0NBpe_tyCUFFX5ieQ/s1600/Road+to+infinity.jpg" height="240" width="320" /></a></div>
<br />
<br />
When you counted them all, one picket two meters high appears. Then again the short ones, start counting again: one, two, three...<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZ0-PTjYnWzJs1Gfyf17lnnqFvddqj8yanNthJyw6s8VjgX8_EsA5r_jm3njEUT_jW_EqLzpWZ15l4mkPHCT7kKZh64_gw-zi-KyoAh4KgVf6y8994wk2OgMRes5F5QIU4N9eFF22Urw/s1600/Road+to+infinity+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZ0-PTjYnWzJs1Gfyf17lnnqFvddqj8yanNthJyw6s8VjgX8_EsA5r_jm3njEUT_jW_EqLzpWZ15l4mkPHCT7kKZh64_gw-zi-KyoAh4KgVf6y8994wk2OgMRes5F5QIU4N9eFF22Urw/s1600/Road+to+infinity+2.jpg" height="240" width="320" /></a></div>
<br />
<br />
counted them all the second picket two meters high appear. Continue like this, and a third, a fourth will appear, and so on. When you counted all the two meters high pickets, one three meters high appears!<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiF0Ovz28ZxHnWJTPrnCQAEJn0OvAJYnL3FRrHn-r90PunlSfhMrc444iy7Qj0hb1QR7uPmDlfmyRTsWvwQ5iMGfXKlULB4YBVbtE8krhIj7RK5MpXoTFNA5dufaRuetA0szMnwBC5bPg/s1600/Road+to+Infinity+3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiF0Ovz28ZxHnWJTPrnCQAEJn0OvAJYnL3FRrHn-r90PunlSfhMrc444iy7Qj0hb1QR7uPmDlfmyRTsWvwQ5iMGfXKlULB4YBVbtE8krhIj7RK5MpXoTFNA5dufaRuetA0szMnwBC5bPg/s1600/Road+to+Infinity+3.jpg" height="240" width="320" /></a></div>
<br />
<br />
And then the small ones again, etc. After the second batch of two meters high there is a second three meters picket, after the third batch a third, and so on. Once all the three meters high are finished, here it is one of four meters. Continue like this, and you will have passed <span style="font-family: Calibri, sans-serif; font-size: 11pt; line-height: 16.866666793823242px;">ω</span><sup style="font-family: Calibri, sans-serif; line-height: 15.333333015441895px;">ω</sup> pickets.<br />
<br />
Let's try even another way. Imagine a tally counter. But instead of having just numbers between 0 and 9, every disk has ALL the numbers. And instead of having finite disks, it has infinite disks. There: the tally counter can count exactly <span style="font-family: Calibri, sans-serif; font-size: 11pt; line-height: 16.866666793823242px;">ω</span><sup style="font-family: Calibri, sans-serif; line-height: 15.333333015441895px;">ω</sup> numbers.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8UWyVy4f8vqLzDj_MoHq6GsFAswsq-TUKBPLbaflPZNKWkJaQ3UQ6j9of5uDmXUF2WwNeEZgB-DTfmc6KmSnXS4nKCp4av7FeZouD1Flt666B_u3ajFL7NHgRmDnxCw4dwGO8Ex5u-w/s1600/Infinity+counter.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8UWyVy4f8vqLzDj_MoHq6GsFAswsq-TUKBPLbaflPZNKWkJaQ3UQ6j9of5uDmXUF2WwNeEZgB-DTfmc6KmSnXS4nKCp4av7FeZouD1Flt666B_u3ajFL7NHgRmDnxCw4dwGO8Ex5u-w/s1600/Infinity+counter.jpg" height="50" width="320" /></a></div>
<br />
Homework! How to visualize <b>Infinity to the power of infinity to the power of infinity</b>? And <b>Infinity to the power of infinity to the power of infinity </b><b>to the power of infinity</b>? And <b>Infinity made to the power of infinity infinite times</b>? Have fun!<br />
<br />
One personal note: it is disarming how practically all the children manage to think about infinity plus one, like above pretty much. <b>And yet many times they are stopped by adults, that tell them that you cannot do infinity plus one, because infinity is infinity. Of course you can!</b> This reminds me of a schoolmate at kindergarten that once told me that 100 is the biggest number. I asked her "What about 103?" and she answered "It does not exist". In the same vein, my teacher at primary school told me that you cannot do 4-6. "Isn't it -2?" I asked. She told me "That does not exist, and that operation is FORBIDDEN". I am still recovering from that. So please, if you are reading this, think of the children! Don't limit their fantasy, don't close them the gate to Cantor's paradise!<br />
<br />
<b>I think that the people that make the most fascinating jobs have still inside them the kid that they were.</b> Every astronaut is driven by the "childish" fascination of the stars, every zoologist by the passion kids have for animals, and what about paleontologists and dinosaurs! Well, <i>si parva licet</i>, I am still playing the game of "who says the biggest number".<br />
<br />
And I am winning.<br />
<br />
<i>The story does not end here. You can find an update at <a href="http://cantorontheshore.blogspot.it/2014/08/infinity-times-infinity-updates.html">this page</a>. Also, did you know that there is a <a href="http://cantorontheshore.blogspot.it/2014/08/this-are-infinites-that-count.html">dance song</a> about all of that?</i>Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com0tag:blogger.com,1999:blog-8155036766532593680.post-43903539363751151872014-08-04T07:53:00.001-07:002015-02-24T08:35:46.422-08:00Don't be naive, Google!And now this blog is as contemporary as ever!<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsJK9q7GvcovFRk_0P1DKbCAf3UQMlzUSQ87hjTthCu6JLUhCIfjnpeecvoSfw1zsIEIWtuHxEx2Y0R6FyGWqgHFGOk-P31glnb-WeaSkaE9mDZscCL1-JXVk0CjwsPwER0wH4UuQT9A/s1600/Doodle+Venn.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsJK9q7GvcovFRk_0P1DKbCAf3UQMlzUSQ87hjTthCu6JLUhCIfjnpeecvoSfw1zsIEIWtuHxEx2Y0R6FyGWqgHFGOk-P31glnb-WeaSkaE9mDZscCL1-JXVk0CjwsPwER0wH4UuQT9A/s1600/Doodle+Venn.png" height="124" width="320" /></a></div>
<br />
Google's <a href="http://www.google.com/doodles/john-venns-180th-birthday" target="_blank">doodle</a> today is about the 180th birthday of John Venn, the "inventor" of <a href="http://en.wikipedia.org/wiki/Venn_diagram" target="_blank">Venn diagrams</a> (their history, in fact, is much more complicated).<br />
<br />
Now, I know I have been a bit harsh in my <a href="http://cantorontheshore.blogspot.co.at/2014/07/this-blog-will-blow-your-mind-aka.html" target="_blank">manifesto</a> about them. I should apologize: they really are important.<br />
<br />
<b>Just think of the concept of set.</b> It's something very extravagant and abstract: it splits the world exactly in two, the things that are in the set, and the things that are not. How could we arrive at that? The real world is much more vague: think for example of the set of chairs. <b>Can you really say which objects are in it and which are not?</b> Is a stone in it? You can sit on it! And a doll chair? You cannot sit on it, why do you call it a chair? Bah!<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjG5xvAX158-_fQCaxPnSQB6ONjbRMkyJWdD_faZ_rAdqljNslwI9mqQxWV_I5ilxEynrq4NVCH4_0Pu-It8PoxulaTPAG0F5CBd26LTnImw-4ejb4jdAgbqwRDq7kewqEPHwJjU_thcA/s1600/mike_womack.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjG5xvAX158-_fQCaxPnSQB6ONjbRMkyJWdD_faZ_rAdqljNslwI9mqQxWV_I5ilxEynrq4NVCH4_0Pu-It8PoxulaTPAG0F5CBd26LTnImw-4ejb4jdAgbqwRDq7kewqEPHwJjU_thcA/s1600/mike_womack.jpg" /></a></div>
<div style="text-align: center;">
<span style="font-size: x-small;">Is this a chair? You tell me.</span></div>
<br />
<br />
I mean, Plato would say that there is an <a href="http://en.wikipedia.org/wiki/Idea" target="_blank">idea</a> of the chair, and then the physical objects adhere more or less to that idea. There is no sharp line between "chairs" and "not chairs".<br />
<br />
Yet <b>a set-theoretic approach is exactly what is needed for the abstract thinking</b>. <b>Every concept entails the existence of the set of objects that apply to that concept</b>, and (here is the master stroke) <b>you can use that set <i>as an object</i>, manipulating it in different ways, creating new concepts even in transcendental ways</b>. One can make intersections, unions, complements, talk about sets of sets or sets of sets of sets. The possibilities are endless. This is what is called <a href="http://en.wikipedia.org/wiki/Naive_set_theory" target="_blank">Naive set theory</a>: not formal, but effective in practice. And the Venn diagrams give an immediate way to imagine it.<br />
<br />
<b>You can find naive set theory everywhere</b>. Think about <b>linguistics</b>: when you say that somebody is beautiful AND smart, you are saying that he is a member of the intersection of the set of beautiful people and of the set of smart people. If you say that tonight you are eating Chinese OR Indian food, you are sayin that what you are going to eat is in the union of the set of Chinese food and the set of Indian food. So AND is in fact an intersection, and OR is a union.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiII4g88iUmK_ySaR1TYsPh6dwAkGdDe8oACff0RsTqX_r2kuyCtPqc8tZVD3OTe3A-dYk8V3OHtPOsfY2qycMuxs7OBhi_t3UzXGcsYp4u0G451m7-wWeAMx5Mi9T_G8sSUiFgiwVmGw/s1600/Venn+diagrams.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiII4g88iUmK_ySaR1TYsPh6dwAkGdDe8oACff0RsTqX_r2kuyCtPqc8tZVD3OTe3A-dYk8V3OHtPOsfY2qycMuxs7OBhi_t3UzXGcsYp4u0G451m7-wWeAMx5Mi9T_G8sSUiFgiwVmGw/s1600/Venn+diagrams.jpg" height="320" width="226" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<br />
But then what about <b>Google search</b>. If you search "Marilyn Manson" "cute bunnies" (the AND is implicit), it (should) give you the intersection of the results for Marilyn Manson and the results for cute bunnies, while if you search for "apocalypse" OR "Facebook down", it will give you the union of the two results. This is clearly just an example of the action of filtering a database, and without it the world as we know it would just collapse.<br />
<br />
Things can get weirder: there is something called <a href="http://en.wikipedia.org/wiki/Set_theory_%28music%29" target="_blank"><b>musical set theory</b></a>. The earliest example I know (but I am not a musicologist, so take it with a grain of salt) is Herma by <a href="http://en.wikipedia.org/wiki/Iannis_Xenakis" target="_blank">Iannis Xenakis</a>, where he established a set of notes and then used set-theoretic operations to change it in different ways.<br />
<br />
In short: yes, <b>Venn diagrams are great</b>. But when you have to do things that are more complicated, they become useless and more formality is needed.<br />
<br />
So the next person that tells me "so you work with Venn diagrams", he's going to feel the intersection of my fist with his mouth.<br />
<br />
<i>A naive approach to sets can lead you to <a href="http://cantorontheshore.blogspot.com/2014/12/teoria-de-conjuntos-poem-for-christmas.html">beautiful poems</a>. But not to <a href="http://cantorontheshore.blogspot.com/2014/10/the-real-meaning-of-to-infinity-and.html">iconic phrases</a>.</i>Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com0tag:blogger.com,1999:blog-8155036766532593680.post-47239451931373045982014-07-28T06:11:00.000-07:002015-02-24T08:31:10.974-08:00A Pint of Aleph<div class="separator" style="clear: both; text-align: center;">
</div>
<div style="margin-left: 1em; margin-right: 1em;">
</div>
<a href="http://cantorontheshore.blogspot.it/2014/07/rendezvous-with-futurama.html" target="_blank">Last week</a> I showed you a frame of Futurama with a peculiar symbol. So, where did I already see this symbol?<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgMWWsgjjJeqL9OPpWWoJOpYEz9ZjMgq1H0sA6iDrSlCjj-qj9mUwDadIzz5wLchZmNnLz-60QJZHpHZvoLrfhF5u4G1BjLv3W262b38gtmVKu_0h7xx36zYgCYh0IKDF0u1xBp_dpuAQ/s1600/ElAleph.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgMWWsgjjJeqL9OPpWWoJOpYEz9ZjMgq1H0sA6iDrSlCjj-qj9mUwDadIzz5wLchZmNnLz-60QJZHpHZvoLrfhF5u4G1BjLv3W262b38gtmVKu_0h7xx36zYgCYh0IKDF0u1xBp_dpuAQ/s1600/ElAleph.jpg" height="320" width="216" /></a></div>
<br />
<b>That letter is the <a href="http://en.wikipedia.org/wiki/Aleph_(Hebrew)" target="_blank">aleph</a>, the first letter of the Hebrew alphabet</b>, that seems to come from a hieroglyph of an ox. In the Kabbalah it refers to the origin of the universe, the "primordial one that contains all numbers". Such fascinating object of course caught the attention of the Argentine writer <a href="http://en.wikipedia.org/wiki/Jorge_Luis_Borges" target="_blank">Jorge Luis Borges</a>, an artist that wrote a lot on the concept of infinity. I can make many posts on his work, but for now let's consider his short story "The Aleph".<br />
<br />
It tells the story of a man who has in his house the Aleph, a point in space that contains all other points. Borges himself sees it and comments it in this way:<br />
<br />
<span style="background-color: white;"><span style="font-family: Courier New, Courier, monospace;">All language is a set of symbols whose use among its speakers assumes a shared past. How, then, can I translate into words the limitless Aleph, which my floundering mind can scarcely encompass?</span></span><br />
<span style="background-color: white; font-family: Times, Helvetica;"><span style="font-size: x-small;">(Translation by <a href="http://www.phinnweb.org/links/literature/borges/aleph.html" target="_blank">Norman Thomas Di Giovanni</a>)</span></span><br />
<br />
This is all very nice, but Futurama's cinema had a 0 under the aleph, right? What does it mean?<br />
<br />
Let's go back to Cantor, our hero. <strike>John Green</strike> He proved that there are <a href="http://cantorontheshore.blogspot.it/2014/07/the-fault-in-our-infinites-part-2.html" target="_blank">infinities bigger than other infinities</a> (in fact, he proved that for any infinity, one can find an infinity that is bigger), but are these infinities ordered? Let me explain:<br />
<br />
suppose that I go to Cantor and I show him Infinity Leopold, then he will provide me Infinity Felix, an infinity bigger than Infinity Leopold. But of course, we can go on, and introduce Infinity Stanislaw, an infinity even bigger, and so on.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiZscOziVMryCmp9udDhQOEbGOq9ZuCVYl3ow4twrDgkuY0MPifI35JBGGYl1rb8d3I50RXSD59v_6N9Z3CUbCWo1gekwSGU571Qb0GsKfI9RhdH3-3MvaVVB5Z6D0is50NLTVT7KgmzA/s1600/Infinities.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiZscOziVMryCmp9udDhQOEbGOq9ZuCVYl3ow4twrDgkuY0MPifI35JBGGYl1rb8d3I50RXSD59v_6N9Z3CUbCWo1gekwSGU571Qb0GsKfI9RhdH3-3MvaVVB5Z6D0is50NLTVT7KgmzA/s1600/Infinities.jpg" height="320" width="320" /></a></div>
<br />
<br />
But now, Infinity Chad comes to disrupt the party with a "Hey, bros". Typical Chad. Where does Chad go? <b>Is it bigger /smaller than the other infinities? Is it something completely different and not comparable (like apples and oranges)?</b><br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWiXz6M7XwCjK7MZtsQE-OMyGC1LffEofUEvU4xTM16O5eQ4p0pL5LyCDGkewn0tI7aOpYEZY9Ept1EjB4pzQZXO-8WdwKlaQFXobe9DckGZ6sn1S2gHLOyhi0tL17EPeQz_1OE_UeTA/s1600/Infinities+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWiXz6M7XwCjK7MZtsQE-OMyGC1LffEofUEvU4xTM16O5eQ4p0pL5LyCDGkewn0tI7aOpYEZY9Ept1EjB4pzQZXO-8WdwKlaQFXobe9DckGZ6sn1S2gHLOyhi0tL17EPeQz_1OE_UeTA/s1600/Infinities+2.jpg" height="320" width="320" /></a></div>
<br />
<br />
Under normal circumstances (*) it turns out that yes, <b>the infinities are linearly ordered, i.e., any infinity is either bigger or smaller than all the others,</b> and Chad (alas) has its place. As there is a limit of the names I can come up with, and nobody wants an Infinity <span style="font-size: x-small;">Iqalussuarniartuqqasaagaluakagunnuuq</span>, we can just call them First Infinity, Second Infinity, and so on. Better! thinks Cantor, let's use a symbol just for that. But <b>which symbol can express the limitlessness of infinity, something that "contains all numbers", something that can translate into words things that a mind can scarcely encompass?</b><br />
<br />
Good catch.<br />
<br />
And so <b>the infinities were called aleph 0, aleph 1, and so on</b>. The smallest one is aleph 0, and it is the infinity of the natural number, the smallest infinity ever. Therefore the aleph 0-plex in Futurama is telling us that in that cinema, for any number, there is a screen. This opens up lots of questions on the physics of that universe, but let's not go there.<br />
<br />
I am cheating a bit, here. Of course, the short story of Borges was written much later than Cantor, so they probably gained both inspiration from the Kabbalah. But we know that Borges was also thinking of Cantor's notation. He writes in a postscriptum:<br />
<br />
<span style="font-family: Courier New, Courier, monospace;"><span style="background-color: white;">For the Kabbala, the letter stands for the </span><i style="background-color: white;">En Soph</i><span style="background-color: white;">, the pure and boundless godhead; it is also said that it takes the shape of a man pointing to both heaven and earth, in order to show that the lower world is the map and mirror of the higher; for Cantor's </span><i style="background-color: white;">Mengenlehre</i><span style="background-color: white;">, it is the symbol of transfinite numbers, of which any part is as great as the whole.</span></span><br />
<br />
Ehm. I don't know how to say this. Who am I to correct the great Borges? Well, but this is the <a href="http://cantorontheshore.blogspot.co.at/2014/07/this-blog-will-blow-your-mind-aka.html" target="_blank">aim of the blog</a>, right? I... I will write it in a smaller text.<br />
<br />
<span style="font-size: x-small;">No! It's not! It is not true that in the transfinite numbers (aka what I called infinities, and now it's called infinite cardinals) ANY part is as great as the whole. Pick a transfinite number, and pick a finite part of it. See? You have a part that is not "as great as the whole". It is true that the infinite is characterized from having parts of it that are as great as the whole (remember <a href="http://cantorontheshore.blogspot.it/2014/07/the-fault-in-our-infinites-part-2.html" target="_blank">0 to 1 and 0 to 2</a>?), but not ANY of them. You screwed up BIG TIME, <i>Borges</i>.</span><br />
<span style="font-size: x-small;"><br /></span>
Phew, I am sorry Borges. But the first quote is <i>so</i> true. <b>Just thinking about infinity, one can feel a sensation that it's not possible to explain in words, something so dizzying that the mind just boggles.</b> I will try <a href="http://cantorontheshore.blogspot.com/2014/08/advanced-thinking-thinking-infinity.html">next time</a> to show you some games that can give you that sensation, with the help of an advertisement and... a kindergarten.<br />
<br />
Stay tuned!<br />
<br />
(*) People think that Mathematics is made of statements that have an universal value, and therefore extremely objective. It is true that mathematical statements have an universal value, but sometimes people don't agree on what the Universe <i>is</i>. There are thus many groups of people inside mathematics, with opinions that differ. For example, <a href="http://cantorontheshore.blogspot.it/2014/07/the-fault-in-fault-in-our-stars-part-1.html" target="_blank">in this post</a> I postulate the existence of infinite sets, therefore going against <a href="http://en.wikipedia.org/wiki/Finitism" target="_blank">finitists</a> (like Kronecker) and <a href="http://en.wikipedia.org/wiki/Ultrafinitism" target="_blank">ultrafinitists</a>. In <a href="http://cantorontheshore.blogspot.it/2014/07/the-fault-in-our-infinites-part-2.html" target="_blank">the next post</a> I define a way to compare infinities that not all people agree with. Well, just one or two don't agree with it, but their results are solid, so it is right to acknowledge them. Finally, for the result above I use the <a href="http://en.wikipedia.org/wiki/Axiom_of_choice" target="_blank">Axiom of Choice</a>, that basically says that if you have infinite pairs of socks, you can pick one for any pair (<a href="http://en.wikiquote.org/wiki/Bertrand_Russell" target="_blank">Russel</a> quote). Here I am going against intuitionists and constructivists. Of course the big majority of mathematicians agrees with me, but at each his own.<br />
<span style="background-color: white; font-family: Times, Helvetica;"><br /></span>
<span style="background-color: white; font-family: Times, Helvetica;"><i>Every problem solved brings with it dozens of new problems to solve. Linearly ordered infinities bring the <a href="http://cantorontheshore.blogspot.com/2015/01/infinite-mess-part-one.html">Continuum Problem</a>, for example. Also, I would say, the aleph would make a <a href="http://cantorontheshore.blogspot.com/2014/12/and-best-tattoo-of-year-2014-is.html">cool tattoo</a>.</i></span>Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com1tag:blogger.com,1999:blog-8155036766532593680.post-31229088476143608872014-07-21T06:11:00.001-07:002015-02-24T03:01:25.056-08:00Rendezvous with Futurama<div class="separator" style="clear: both; text-align: center;">
</div>
<div style="margin-left: 1em; margin-right: 1em;">
</div>
<i>Disclaimer: Very short post, today, since I am recovering from the <a href="http://vsl2014.at/" target="_blank">Vienna Summer of Logic</a> and I feel a bit dizzy from all that logic.</i><br />
<i><br /></i>
Enough teenager love stories! I really overindulged on that. It's a bit creepy. Let's see something more classical:<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://upload.wikimedia.org/wikipedia/en/thumb/d/de/Futurama_title_screen.jpg/265px-Futurama_title_screen.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="The opening title card for Futurama" border="0" data-file-height="270" data-file-width="360" src="http://upload.wikimedia.org/wikipedia/en/thumb/d/de/Futurama_title_screen.jpg/265px-Futurama_title_screen.jpg" height="199" srcset="//upload.wikimedia.org/wikipedia/en/d/de/Futurama_title_screen.jpg 1.5x, //upload.wikimedia.org/wikipedia/en/d/de/Futurama_title_screen.jpg 2x" width="265" /></a></div>
<br />
Futurama is an animated TV show that was advertised at the beginning as "The Simpsons in the future", but was really a brilliant, humorus, post-modern cartoon. It was cancelled, revived again and put on hold another time. The authors were never afraid of going too deep into complex/nerdy matters, like physics or computer science, thus including mathematical subjects (most famously, a writer <a href="http://en.wikipedia.org/wiki/The_Prisoner_of_Benda#The_theorem" target="_blank">invented and published</a> a new math theorem just for an episode), so it is quite expected to find a Set Theory joke. Where? Here it is, from the 21st episode (it appeared also on the 47th episode):<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="http://theinfosphere.org/images/4/46/Number_9_Raging_Bender.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img alt="File:Number 9 Raging Bender.jpg" border="0" src="http://theinfosphere.org/images/4/46/Number_9_Raging_Bender.jpg" height="301" width="400" /></a></div>
<br />
This is in fact pretty popular, and not obscure at all, but if you don't know what I am talking about, you would ask: "What is this? <b>Where is the infinite here?</b> There is a strange letter there, but what has this to do with Set Theory? Are you kidding me? You're wasting my time!"<br />
<br />
To you, I'll tell you two things<br />
<br />
1. I... I don't think this is how the Internet works. I am sorry. Please stop shouting at your screen, and in case write some comment below?<br />
2. I am not going to answer this now, <b>wait until next week</b>, and I will ask help to a key figure in Spanish language literature...<br />
<br />
So, see you next Monday!<br />
<br />
<i>Update: the answer is <a href="http://cantorontheshore.blogspot.it/2014/07/a-pint-of-aleph.html">here</a>, as well as Infinity Chad. And if you like Sci-Fi, maybe you'll want to know about <a href="http://cantorontheshore.blogspot.com/2014/09/artemis-fowl-and-large-cardinals.html">this</a>.</i>Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com0tag:blogger.com,1999:blog-8155036766532593680.post-41870370650572633242014-07-14T10:40:00.002-07:002015-02-21T12:14:47.433-08:00The Fault in Our Infinites (Part 2)<div style="border-bottom: solid windowtext 1.0pt; border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-element: para-border-div; padding: 0cm 0cm 31.0pt 0cm;">
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<span lang="EN-US">Previously on <a href="http://cantorontheshore.blogspot.co.at/2014/07/the-fault-in-fault-in-our-stars-part-1.html">Cantor on the Shore</a>…<o:p></o:p></span></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<span lang="EN-US"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBbVY6glkZrcP-_CtUn3Nfpo0s2DPi-U8bmkTHZiasEqrrQ8ZgFV2z_CtuPm8KEcudZXFdQ_GEJ2__MmSSbLvGTUwa9HktIuJCNB6r84gmB3p2HSrFmJshts2Eb-2VrsFGaUnKyVn1vw/s1600/The_Fault_in_Our_Stars+Reloaded.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBbVY6glkZrcP-_CtUn3Nfpo0s2DPi-U8bmkTHZiasEqrrQ8ZgFV2z_CtuPm8KEcudZXFdQ_GEJ2__MmSSbLvGTUwa9HktIuJCNB6r84gmB3p2HSrFmJshts2Eb-2VrsFGaUnKyVn1vw/s1600/The_Fault_in_Our_Stars+Reloaded.jpg" height="320" width="219" /></a></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<span lang="EN-US">It turns out that in the book “The Fault in Our
Stars”, by John Green, there is a reference on Cantor’s most famous result:
there are infinities bigger than other infinities. The reference was slightly wrong, but
nonetheless alluring, and the message left a seed in the mind of one of the
protagonists (no spoilers!) How did the seed developed? Let’s hear:<o:p></o:p></span></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<span lang="EN-US"><span style="font-family: Courier New, Courier, monospace;">I am not a mathematician, but I know this:
There are infinite numbers between 0 and 1. There’s .1 and .12 and .112 and an
infinite collection of others. Of course, there is a bigger infinite set of
numbers between 0 and 2, or between 0 and a million. Some infinities are bigger
than other infinities. A writer we used to like taught us that. There are days,
many of them, when I resent the size of my unbounded set.</span><o:p></o:p></span></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<span lang="EN-US">NOOOOOOO! <b>This is so wrong!</b> Completely wrong!
Abhorrently wrong. Wrong like socks and sandals. Like ketchup on pizza. Like
“Smells Like Teen Spirit” on the <a href="http://www.youtube.com/results?search_query=smells+like+teen+spirit+ukulele">ukulele</a>. Like </span><span style="text-align: justify;">snow in August (in the northern emisphere). Like w</span><span style="text-align: justify;">hen two people say goodbye and then they walk in the same direction. </span>When I heard this at the cinema I unleashed all my
rightful fury (with the dismay of my fellow moviegoers. Sorry).</div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<span lang="EN-US">To be honest, John Green <a href="http://johngreenbooks.com/questions-about-the-fault-in-our-stars-spoilers/#hazel">claims</a> that he knew it
was wrong (attention, spoilers in the link), but he wrote it wrong on purpose, to show that teenagers can and do
reach incorrect conclusions, but find comfort in them anyway. Yet, there is no indication
in the book (or in the movie) that it is wrong, so it’s easy to imagine a girl
reading the sentence, thinking “There are more numbers between 0 and 2 than
between 0 and 1? Duh, of course. And this is what Cantor said? That Cantor guy
is useless”, then the girl becomes a researcher in the University, they ask her
to evaluate a proposal for a grant, she reads “Cantor” and think “this research
is nothing worthy”, and then I become unemployed. And nobody wants that.</span><br />
<br />
<span lang="EN-US"><o:p></o:p></span></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<span lang="EN-US">So, <b>why there is the same amount of numbers
between 0 and 1 and between 0 and 2</b>?<o:p></o:p></span></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<span lang="EN-US">First, let’s agree on the meaning of “same
amount”, with an example. Let’s go against the timeless suggestions of my
mother, and compare apples and oranges.<o:p></o:p></span></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<span lang="EN-US"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiO8kGlkGmpwrbbwqUPfeLd5Q4d0_lvls6dBQshczBaArovKyGsKXFiCnr_rSsiMhl-zJDUlDiyYLbH8mtW4jcVyvGCIENp-kHcvBR6y8glGRROjk8VJ4IMSp0gCxnPxPyYP1RTOcCKRA/s1600/Before+comparison.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiO8kGlkGmpwrbbwqUPfeLd5Q4d0_lvls6dBQshczBaArovKyGsKXFiCnr_rSsiMhl-zJDUlDiyYLbH8mtW4jcVyvGCIENp-kHcvBR6y8glGRROjk8VJ4IMSp0gCxnPxPyYP1RTOcCKRA/s1600/Before+comparison.jpg" height="213" width="320" /></a></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<span lang="EN-US">If Cantor has a bowl of apples and Kronecker, his
eternal nemesis, has a bowl of oranges, how can they know who has more, since
they don’t trust the counting capabilities of each other?<o:p></o:p></span></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<span lang="EN-US">They pick some string, and start connecting
every apple to one orange, one by one. If in the end the bowls are empty, there
was the “same amount” of oranges, of course. (*)<o:p></o:p></span></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<span lang="EN-US"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgF0AYmo4XJQ69wUz98-VadLG_zDpll0Wcqcyhp8_A7m6xGNwsoyUy-N78SCmwFdyTsoQp8vfqfnukNgiIYnTLRCOkka0FDU0LVxlYAqvnHQU6IdHJvHwQf0xFuzUj1o6srd3rcuss8uQ/s1600/Aftercomparison.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgF0AYmo4XJQ69wUz98-VadLG_zDpll0Wcqcyhp8_A7m6xGNwsoyUy-N78SCmwFdyTsoQp8vfqfnukNgiIYnTLRCOkka0FDU0LVxlYAqvnHQU6IdHJvHwQf0xFuzUj1o6srd3rcuss8uQ/s1600/Aftercomparison.jpg" height="213" width="320" /></a></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<span lang="EN-US">And this is what we are going to do with the infinities.
<b>If I can connect any number between 0
and 1 to a number between 0 and 2, finally without leaving out numbers, it is
clear that there is the same amount of numbers.</b> Here we go: connect every
number between 0 and 1 with its double, like this<o:p></o:p></span></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<span lang="EN-US"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglDX8HzWuRIAbfn49ngZNrn3VszbbutETitfdrSDMeR1e1oi9ehz1o9LyG8-AkiSjpEeRzuOeDyT5TF3zXR3M5YpnfnVJHfUe2kTmlsiCmcpO45rPj0WlG32dVZ_tXvr4Qmg7rkrV_hA/s1600/Third+Comparison.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglDX8HzWuRIAbfn49ngZNrn3VszbbutETitfdrSDMeR1e1oi9ehz1o9LyG8-AkiSjpEeRzuOeDyT5TF3zXR3M5YpnfnVJHfUe2kTmlsiCmcpO45rPj0WlG32dVZ_tXvr4Qmg7rkrV_hA/s1600/Third+Comparison.jpg" height="213" width="320" /></a></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<br /></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm;">
<br /></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm; text-align: justify; text-justify: inter-ideograph;">
<span lang="EN-US">And so on. <b>Did we leave out some number? Nope.</b> Every number between 0
and 2 is connected with its half, so 0.3 is connected with 0.15, 1.33 is
connected with 0.665 and so on.<o:p></o:p></span></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm; text-align: justify; text-justify: inter-ideograph;">
<span lang="EN-US">Once you see it, it is pretty clear (even for a teenager? Yes? Please
tell me so, I have such a good faith in teenagers). So that’s it: I righted the first error.<o:p></o:p></span></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm; text-align: justify; text-justify: inter-ideograph;">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm; text-align: justify; text-justify: inter-ideograph;">
<span lang="EN-US">Yes, because there is another mistake! The parade of the errors seems to
have no end. Maybe. I don’t know, the problem is that I simply don’t understand
the last sentence. It’s like many pretty words with no real meaning. So, since
probably John Green won’t come here to explain this, I have to guess, and it
seems to me that the protagonist is lamenting that there are too many numbers
between 0 and whatever. But then the word “unbounded” is wrong! That set is not
unbounded, it is perfectly bounded, by whatever. But it’s infinite. Yep,
unbounded and infinite are two different things.<o:p></o:p></span></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm; text-align: justify; text-justify: inter-ideograph;">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal" style="border: none; mso-border-bottom-alt: solid windowtext .75pt; mso-padding-alt: 0cm 0cm 31.0pt 0cm; padding: 0cm; text-align: justify; text-justify: inter-ideograph;">
<span lang="EN-US">Another thing! Like Columbo, I want to add just a small thing before
leaving: even if the premises are wrong, in the end the protagonist is right in
finding comfort in the infinite. <b>There <i>is</i> an infinity of moments even in a
short period of days, so you can project an eternity in there.</b> So romantic
(and, most importantly, correct)!</span><br />
<span lang="EN-US"></span><br />
<span lang="EN-US"><o:p><br /></o:p></span>
<span lang="EN-US"><o:p>(*) One can, in fact, refuse to use this as a method of measuring the "same amount". But then, what can we use? Some mathematicians came out with a definition such that, yes, there are more numbers between 0 and 2, but the definition is incredibly complex and unintuitive. I am open to the possibility that in the future there will be an intuitive definition that satisfy our intuition, but until then, we do with what we have. If you have better ideas, please write them in the comments!</o:p></span><br />
<br />
<span lang="EN-US"><o:p><i>Literature is in fact a great resource for infinity. You enjoyed apple and oranges? <a href="http://cantorontheshore.blogspot.com/2015/01/infinite-mess-part-two.html">Here</a> is something even more challenging, courtesy of David Foster Wallace. </i></o:p></span><i>And if all this infinities counfuse you, <a href="http://cantorontheshore.blogspot.com/2014/07/a-pint-of-aleph.html">Borges</a> has a nice way to put some order.</i><br />
<span lang="EN-US"><o:p><i><br /></i></o:p></span></div>
</div>
Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com1tag:blogger.com,1999:blog-8155036766532593680.post-11669251982006384112014-07-07T00:22:00.001-07:002017-06-20T03:03:45.699-07:00The Fault in the Fault in Our Stars (Part 1)<div class="MsoNormal">
<span lang="EN-US">Science
fiction and ”serious and intellectual” novels sometimes contain references to
Set Theory, but starting the blog with them would be cheap. Of course there are
references, these writers are geeks, what do you expect? I want to begin with a
blast! Somewhere unexpected! So, what about… a young adult romance novel?<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfZr_Qnrkm0Pv1JEl4NnTdCKO3ZSrJHxDfxFlguTyCPGXzRh9PMoWeymba18Fgjj67I3QVkFCD1m9FXVeHz6ZBAVtFbWuo_wKxQ2JQkzIHN3wg7MZD7jAiU6liy39yjuMkPwAabtYTrQ/s1600/The_Fault_in_Our_Stars.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhfZr_Qnrkm0Pv1JEl4NnTdCKO3ZSrJHxDfxFlguTyCPGXzRh9PMoWeymba18Fgjj67I3QVkFCD1m9FXVeHz6ZBAVtFbWuo_wKxQ2JQkzIHN3wg7MZD7jAiU6liy39yjuMkPwAabtYTrQ/s1600/The_Fault_in_Our_Stars.jpg" width="219" /></a></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US">John Green
is a popular writer of young adults novels, mostly heartbreaking teen romances,
and he’s pretty good at it. If you like the genre, it doesn’t get any better
than that. Despite the constraints of the genre, he always tries to put
something new in its stories. My
attention goes to “The Fault in Our
Stars”, a novel published in 2012, and then adapted in film in 2014. And yes,
he went there. This is an excerpt:<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<span style="font-family: "courier new" , "courier" , monospace;"><span lang="EN-US"> “So Zeno is most famous for his tortoise
paradox. Let us imagine that you are in a race with a tortoise. The tortoise
has a ten-yard head start. In the time it takes you to run that ten yards, the tortoise
has maybe moved one yard. And then in the time it takes you to make up that
distance, the tortoise goes a bit farther, and so on forever. You are faster
than the tortoise but you can never catch him; you can only decrease his lead.<o:p></o:p></span></span></div>
<div class="MsoNormal">
<span style="font-family: "courier new" , "courier" , monospace;"><span lang="JA" style="mso-ansi-language: EN-US; mso-ascii-font-family: Calibri; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: "MS 明朝"; mso-fareast-theme-font: minor-fareast; mso-hansi-font-family: Calibri; mso-hansi-theme-font: minor-latin;">“</span></span><span lang="EN-US"><span style="font-family: "courier new" , "courier" , monospace;">Of course, you just run past the tortoise without contemplating the
mechanics involved, but the question of how you are able to do this turns out
to be incredibly complicated, and no one really solved it until Cantor showed
us that some infinities are bigger than other infinities.”</span><o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US">So, is this
correct or not?<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US"><b>It goes
very, very close</b>, I am tempted to say that is correct, but <b>no, it isn’t</b>.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US">Let’s start
from the beginning: Zeno’s paradox of <a href="http://en.wikipedia.org/wiki/Achilles_and_the_Tortoise#Achilles_and_the_tortoise">Achilles and the Tortoise</a>. The quote does
a good job to illustrate it, down to the big problem. <b>Why “you can never catch
him”?</b> Because it would take an infinite amount of intervals of time? Or of
space? Because the sequence never ends? </span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US">Anyway, <b>all of this was solved in the
19<sup>th</sup> century</b>, during the complete overhaul of Analysis that has been
made by, for example, <a href="http://en.wikipedia.org/wiki/%CE%95-%CE%B4_definition#Weierstrass_definition_.28epsilon-delta.29_of_continuous_functions">Weierstrass</a> and Cauchy: it simply is possible to have a
sum of infinite terms that gives a finite result. So Achilles will catch the elusive tortoise in finite time (or space).<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US">Problem
solved? Nuh-uh. You wish. Yes, <b>the solution made perfect sense and the
calculations worked, but it was not sound, because everybody was scared by
infinite sums</b>. “Infinity is just for God! How can we puny humans deal with
this!” </span><br />
<span lang="EN-US"><br /></span>
<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhzOMnQ7U-CPwyXoiLt7_22MH6dx09nOcKPteNy9Ny_qoK-7GPD-R7LGidw9UvKQNQMpN0xONhKwCST0w4r3rDbNgxaA8LbVQdzMqqnmk-G1Ad1DeUQgO5LwGrCZVsV09RGSqyXUbSZFA/s1600/PanicKronecker.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhzOMnQ7U-CPwyXoiLt7_22MH6dx09nOcKPteNy9Ny_qoK-7GPD-R7LGidw9UvKQNQMpN0xONhKwCST0w4r3rDbNgxaA8LbVQdzMqqnmk-G1Ad1DeUQgO5LwGrCZVsV09RGSqyXUbSZFA/s1600/PanicKronecker.gif" width="320" /></a></div>
<div style="text-align: center;">
<span style="font-size: x-small;">(Imagine <a href="http://en.wikipedia.org/wiki/Leopold_kronecker">Kronecker</a> with a shrill voice saying this)</span></div>
<span lang="EN-US"><br /></span>
<span lang="EN-US">It was Aristotles
fault, of course, like it always is: he practically said that the only way to
think about infinity is to think of finite sets larger and larger, but we
cannot deal with an infinite set. <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US">And here
comes our hero Cantor to save us, daiquiri in hand! With his study,<b> he proved
that it was mathematically sound to deal
with infinite sets</b>, therefore giving the theoretical support necessary to
Analysis to solve Zeno’s paradox. I am not saying this, Russell <a href="http://books.google.at/books?id=PB8I0kHeKy4C&pg=PA319&lpg=PA319&dq=russell+cantor+solved+zeno%27s+paradox&source=bl&ots=1iPbT4z2gZ&sig=y0fVZ9FNX0_zO7eujwRLdpnXFRE&hl=it&sa=X&ei=brG5U9DZBczH7AaH04HQBQ&redir_esc=y#v=onepage&q=russell%20cantor%20solved%20zeno%27s%20paradox&f=false">said it</a>. Also, at the same time with his brilliant work he canceled centuries of superstition, advanced the knowledge of the entire humanity and validated all the smart kids that to win at the game of the biggest number shouted "<a href="http://en.wikipedia.org/wiki/Infinity_plus_one">Infinity plus one</a>!"<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US">So, Zeno
-> Cantor, right? Where is the error? Well, John Green quotes Cantor’s
Theorem: yes, even if it goes against our
intuition, there are infinities bigger than other infinities. This is the
starting point of modern Set Theory, the pedestal upon which the magnificent
crystal castle is built. The proof of this is practically everywhere, it is
called <a href="http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument">Cantor’s Diagonalization</a>. But! <b>It is not the solution to Zeno’s paradox.
His treatment of infinity (that lead him to his Theorem) is.</b><o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
</div>
<div class="MsoNormal">
<span lang="EN-US">Ok, I admit
it, that was very nitpicky. I forgive John Green, because he couldn’t write the
history of 19<sup>th</sup> century mathematics in one line, right? But I won’t forgive
what he did next, because, believe it or not, that was not the only occurrence
of infinity in the book. Something much worse is coming…<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US"><br /></span></div>
<div class="MsoNormal">
<span lang="EN-US">Stay tuned for the <a href="http://cantorontheshore.blogspot.it/2014/07/the-fault-in-our-infinites-part-2.html">next episode</a>!</span><br />
<span lang="EN-US"><br /></span>
<span lang="EN-US"><i>Update: in another post I explain in more details why Cantor's approach solves the tortoise paradox. If you are unsatisfied by the explanation above, go <a href="http://cantorontheshore.blogspot.co.at/2014/10/the-real-meaning-of-to-infinity-and.html">here</a>. And there is also Buzz Lightyear! But I stress: Cantor's Theorem has nothing to do with it.</i></span><br />
<span lang="EN-US"><i>You're wondering why John Green made this error? The behind-the-scenes answer is <a href="http://cantorontheshore.blogspot.com/2015/01/infinite-mess-part-two.html">here</a>!</i></span></div>
Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com0tag:blogger.com,1999:blog-8155036766532593680.post-90510588798657216422014-07-02T11:48:00.000-07:002014-07-02T11:48:51.080-07:00This Blog Will Blow Your Mind! (a.k.a. Mission Statement)<div class="MsoNormal">
<a href="https://www.blogger.com/blogger.g?blogID=8155036766532593680" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><span lang="EN-US">Hello
everyone, my name is Vincenzo Dimonte and I am a researcher in Set Theory (as
long as the contract holds, so not for very long). In other words, I spend my
time writing applications for grants, project proposals, CV and so on, and in
the spare time I do some research in Set Theory. “But come on!” I hear you
blurt. “What there is to research in sets?
Aren’t they just colored potatoes?”<o:p></o:p></span></div>
<table align="left" cellpadding="0" cellspacing="0">
<tbody>
<tr>
<td height="13" width="52"></td>
</tr>
<tr>
<td></td>
<td></td>
</tr>
</tbody></table>
<div class="MsoNormal">
<!--[if gte vml 1]><v:oval id="Ovale_x0020_1" o:spid="_x0000_s1026"
style='position:absolute;margin-left:40.2pt;margin-top:10.8pt;width:114.75pt;
height:64.5pt;z-index:251659264;visibility:visible;mso-wrap-style:square;
mso-height-percent:0;mso-wrap-distance-left:9pt;mso-wrap-distance-top:0;
mso-wrap-distance-right:9pt;mso-wrap-distance-bottom:0;
mso-position-horizontal:absolute;mso-position-horizontal-relative:text;
mso-position-vertical:absolute;mso-position-vertical-relative:text;
mso-height-percent:0;mso-height-relative:margin;v-text-anchor:middle'
o:gfxdata="UEsDBBQABgAIAAAAIQC75UiUBQEAAB4CAAATAAAAW0NvbnRlbnRfVHlwZXNdLnhtbKSRvU7DMBSF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" filled="f" strokecolor="#243f60 [1604]" strokeweight="2pt"/><v:oval id="Ovale_x0020_2"
o:spid="_x0000_s1027" style='position:absolute;margin-left:129.45pt;
margin-top:10.8pt;width:102pt;height:64.5pt;z-index:251660288;visibility:visible;
mso-wrap-style:square;mso-wrap-distance-left:9pt;mso-wrap-distance-top:0;
mso-wrap-distance-right:9pt;mso-wrap-distance-bottom:0;
mso-position-horizontal:absolute;mso-position-horizontal-relative:text;
mso-position-vertical:absolute;mso-position-vertical-relative:text;
v-text-anchor:middle' o:gfxdata="UEsDBBQABgAIAAAAIQC75UiUBQEAAB4CAAATAAAAW0NvbnRlbnRfVHlwZXNdLnhtbKSRvU7DMBSF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" filled="f" strokecolor="#f79646 [3209]" strokeweight="2pt"/><![endif]--><!--[if !vml]-->
<!--[endif]--><span lang="EN-US"> </span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEioP_ugKkhHwc6TVjdkc0d4ahmgXuPpGZCA6iHPNMWQimL55hR335ynjyvj2eYZPZA_hDf4k6R-sjVlwvj_FOWUi8N5uhrmHHlskMCK5fWKYJg-yRlHdBXLRCDQxtCYYAgmGZErgo6b_Q/s1600/potatoes.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEioP_ugKkhHwc6TVjdkc0d4ahmgXuPpGZCA6iHPNMWQimL55hR335ynjyvj2eYZPZA_hDf4k6R-sjVlwvj_FOWUi8N5uhrmHHlskMCK5fWKYJg-yRlHdBXLRCDQxtCYYAgmGZErgo6b_Q/s1600/potatoes.jpg" /></a></div>
<br /></div>
<div class="separator" style="clear: both; text-align: center;">
</div>
<div style="text-align: center;">
<span style="font-size: x-small;">Above: Set Theory?</span></div>
<br />
<div class="MsoNormal">
<span lang="EN-US"> <o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">Well, that
is the starting point, just like learning to write the letter A is the starting
point for literature.<o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: center;">
<span style="font-family: "Comic Sans MS"; font-size: 26.0pt; line-height: 115%;"><br /></span>
<span style="font-family: "Comic Sans MS"; font-size: 26.0pt; line-height: 115%;">A a B b C c D d E e</span><br />
<span style="font-size: x-small;"><br /></span>
<span style="font-size: x-small;">Above: Literature?</span></div>
<div class="MsoNormal">
<div style="text-align: center;">
<o:p></o:p></div>
</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-US">Set Theory
grew very much from its roots, and now it is one of the most complex and
difficult fields of Mathematics, capable of giving headaches even to the most
skilled mathematician. And I work on that! Amazing.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">Where does
the difficulty come out? Infinity. As long as the sets are finite, things are
pretty much what we expect (not trivial, though, it is incredible the amount of
work we can do with finite sets), but when infinities come into play, then all
the world as we know goes upside down.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">Anyway, as
humanity has been fascinated by infinity since the dawn of time, it happens
that Set Theory creeps in popular medias: books, movies, etc. But is it done
correctly? How are writers dealing with such a difficult topic? This is why
this blog exists: it is a collection of instances of Set Theory in pop culture,
with comments on whether the quotation has been done surprisingly well, or
disastrously wrong.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-US">I suspect I
will update sparsely: I will just post anytime I find something. So, if you are
reading/watching/hearing something about infinity and you wonder if it is
correct, comment here, or contact me! I will solve your doubt.<o:p></o:p></span></div>
<br />
<div class="MsoNormal">
<span lang="EN-US">One last
thing: the title of the blog. The impulse for its creation comes from the
belief that there is no contest between science and humanities, both concur in
the progress of mankind, and yet both have a lot to learn from each other. This
is a very metamodernist concept, and for that reason the title of the blog is a
variation of “Kafka on the Shore”, the
book from the most metamodernist author I know, Haruki Murakami. Also, I
enjoyed the view of Cantor, the founder of Set Theory, chilling on the shore
with a daiquiri in his hand, and solving doubts in his spare time.<o:p></o:p></span></div>
Vincenzo Dimontehttp://www.blogger.com/profile/18171790656280332547noreply@blogger.com2