Tuesday, December 30, 2014

And 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?

Yet, Buzzfeed made a list of the most trendy tattoos of the year 2014. And what do we have here?

Photo from here

The lemniscate really exploded this year. 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 Buzz Lightyear reminder!)

Photo from here

and the combos "feather+infinity" and "finger tattoo+infinity" are also ubiquitous.

Photo from here

Photo from here

There are tumblrs devoted to it and all sorts of websites. But I am a set theorists, not a fashion designer, so what can I say to you about it?

Well, now I am about to dump to you a nugget of information that will surely make you popular with your friends. You can ask: why there is one symbol of infinity, when there are many infinities? The reason is that the lemniscate, in mathematics, always refers to the potential infinity, i.e., the infinity you cannot reach, (I explain this here) while each different actual infinity, i.e., an infinity considered as size of an infinite set, has a different symbol. Cool, huh?

Pick up artist suggestion
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".

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.

You want a symbol that it's even more than infinity? Here you are

Monday, December 22, 2014

Teoría de conjuntos (a poem for Christmas)

Christmas is back! The most... depressing period of the year. Really, while the peak of suicides over Christmas is a myth, 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), creepy 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 seizure. And what about the perfect present you have to choose for your Nazi sympathizer cousin that you managed to avoid for one year?

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"

Teoría de conjuntos

su armonía y
su desarmonía.
En algunos casos
la suma de armonías
puede ser casi
empalagosa.
En otros
el conjunto
de desarmonías
produce algo mejor
que la belleza.

(taken from here)

This is a poem by Mario Benedetti, 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.

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.

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?

I'll take Benedetti's word here.

Thanks to Luz for having pointed me out this poem! See? The suggestions box works!

Looking for something light? Then naive Set Theory is for you.

Monday, November 3, 2014

Intermission

It 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, Hello Kitty Hell, would you?

So, today no pop culture. Instead here is a video:

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.

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).

Sit back and enjoy the ride!

Here is some more connection between Set Theory and Theology. Oh, and the above is divulgation done right. Wanna see divulgation done wrong?

Tuesday, October 21, 2014

The 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.

To... boundlessness and... above?

I'm talking of course about the infamous catchphrase of Buzz Lightyear in Toy Story or, as Imdb puts it:

[repeated line
Buzz: To infinity, and beyond!

What is the meaning of "To infinity and beyond?"

It seems a throwaway line, but hidden there is a deep philosophical meaning. It pops up here and there in pop culture and even in random discussions with friends. In 2001: A Space Odissey just before the psychedelic "Star Gate" sequence a title card appear, "Jupiter and Beyond the Infinite" (an inspiration for Buzz?). In the hit Single Ladies (Put a Ring on It), Beyoncé sings "... and delivers me to a destiny, to infinity and beyond.", There are then the album Beyond the Infinite, by trance group Juno Reactor, or the Greek film Eternity and a Day, by director Theo Angelopoulos. In 2008 this sentence helped father and son to survive a shipwreck. And let's not forget infinity plus one! And the lovey-dovey couples smooching "I love you infinity times and more". There are probably infinite other examples (and more!).

It sounds paradoxical: how can you go beyond the infinity? 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 potential infinity to actual infinity.

Let me go back to the example of Achilles and the tortoise, 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?

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 end 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.

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 actual infinite. You pick all the infinity, you put it in a sack, and you treat it as a whole, completed object. So you can imagine what happens after it, in this case Achilles overtake the tortoise, turns and makes very funny faces. "Brlbrblrblrl".

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 you can embrace everything with a single view. (Of course, the metaphor works only if our Earth is flat and infinite, sorry, so I guess that makes no sense).

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:

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:

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!

Artist's rendition

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!

You know what? Let me see what the Internet thinks that "To infinity and beyond" means...

Oh, boy.

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.

This is the first results in Google (hopefully the second after I post this).

Definition in Mathematical Circles:
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."

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:

The asymptote is that dashed line. See? The function crosses it. Here is a vertical asymptote:

The function doesn't cross the line. And Buzz can! 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?

INFINITY!

So, forget about what you read online. The meaning of "To infinity and beyond", even mathematically, is: 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.

You hear that, potentialists?

PS. I am actually going to Harvard next week. If I meet Dr. Sanjay Gupta, I'll let you know.

Statistics says that probably this is the first page of the blog you are reading. Then, may I suggest to read the manifesto, to understand what is this about? Or just skip to the meaty parts, like the posts on "The Fault in Our Stars", or "The Big Bang Theory", or a cute AT&T advertisement.

Monday, October 6, 2014

One Reinhardt and counting...

The original post has been changed: the attribution of the game has been clarified.

Can you hear it? Do you hear that sound? It's something like streeeeeeeeaaaaaaatch. It's me stretching the theme of the blog to accomodate this topic.

You know 2048? Of course, everybody knows it. It's a very addicting game by Gabriele Cirulli, and the Wall Street Journal called it "almost Candy Crush for math geeks".

I cannot describe it.  Just play it. Or don't! You have to read this blog, don't get distracted.

The game is open source, 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?

If you dare to adventure the deep, deep academia web you'll find a specific version... with large cardinals! Now also you can climb up the large cardinals hierarchy! The name of the game is Reinhardt. The original idea was by Yizheng, and then it was picked up by Chris La Sueur, that changed the large cardinals and published it.

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 1=0, and it is cleaner and more strategical.

I thought about giving you the list of the cardinals involved, but why ruining the fun? Part of the delight is in the exploration!

But why it is called Reinhardt? This is the skeleton in the closet of all set theorists.

The reaction of set theorists when asked about Reinhardt and large cardinals

Once upon time (1969, to be precise), William Reinhardt was studying some large cardinal, and had an idea: to build the largest cardinal ever, the king of large cardinals, the most powerful! We now call it Reinhardt cardinal.

But just after months, Kenneth Kunen proved that it was too big and powerful. In fact it just couldn't exist! 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, Reinhardt cardinal is the largest cardinal, even if it does not exists, and the aim of the game is to reach it.

And why the big one is called 1=0? This is more difficult to say, probably I will write it in another post. (Edit: here it is!)

Now you can go playing! Have a nice waste of time!

Many large cardinals that appear in the game already appeared here. Interested in just one large cardinal? Make it this one, says Sheldon.

Monday, September 29, 2014

God and the Big Bang Theory

The 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.

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.

Until this week! When this discussion 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

Oh, I'm working on my lesson plan for Wolowitz. He is going to be so lost. Look at this section over here.

Even I don't really understand it.

I DO! I DO! You suck, Sheldon! Ha! Let's see it closer:

This is the famous proof by Gödel of the existence of God (better known as Gödel's ontological proof). Woah. I will leave you a moment to contemplate the magnificence of this discovery.

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 logical (gah). Why do you always have to fight? He didn't really proved the existence of God, in reality. The right way to see it is as part of the history of philosophical logic.

It was pretty common in the old days to try to prove the existence of God via logical arguments. Think Anselm of Canterbury, Descartes, guys like that. Gödel found logical mistakes in those arguments, and just rewrote them in the current logic language, i.e. modal logic, 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 real. It is just syntax, empty words, he was interested in the reasoning, not in its connections with reality.

Basically (I am simplifying here) Gödel said:

A1) a positive property can have only positive properties as consequences (optimist)
A2) a property is positive if and only if its negation is not positive
A3) being God is positive (this is slightly an understatement)
A4) if a property is positive, it is because it must be positive (nothing left to chance here)
A5) necessary existence is a positive property (again, quite optimist, if something must exists, then it is good)

If 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.

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.

This is what Sheldon wrote, translated from formalese to English:

A2) a property is positive if and only if its negation is positive

But Sheldon! What did you write?! How can this be? You really didn't understand the proof.

And what about Set Theory? This is where things got interesting. I told you that God exists if  there exist positive properties. Do they? Harvey Friedman has a manuscript on that, where he defines God and positive properties more carefully, and proves that they exist if... if...

Ta-da!

if there exists a measurable cardinal! Step aside, Pope. We did your job. Talking about solving unsolvable problems with large cardinals!

Update: it turns out that Gödel hit a wall here. In 2013 someone managed to prove (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.

You want to build a measurable y yourself? Go here. You like to see discussions about infinity between theologians and set theorists? Then this page is just for you.

Monday, September 22, 2014

Artemis Fowl and the Large Cardinals Paradox (part two)

This post continues from here.

this sentence in particular:

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.

and we were trying to understand what it means "solving unprovable theorems with large cardinals". In the previous post I introduced large cardinals (seriously, if you haven't read it, do it now, otherwise it's going to be hard). Now we go back in time, to the beginning of the 20th century, deep in the paradox.

Mathematics, like art, politics, physics, was in a horrible crisis. There was the legitimate doubt that all mathematics was in fact a house of cards. Everything could collapse any minute: people just realized that the math building had no foundation. 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 Isis, and he was talking about recliners (you've got to chill out, dude, it's ok to hate them but not that 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.

Yep, mathematics was in a middle age crisis.

It works like this:

What is a triangle? A polygon with three edges and three vertices.

What is a polygon? A plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed chain or circuit.

What is a plane? A flat, two-dimensional surface.

What is a dimension of an object? The minimum number of coordinates needed to specify any point within it.

What is a point? Ehm... something with no dimensions? No, we defined dimension using points. Then... What?

You can understand how, in a subject like mathematics that strives to be as objective as possible, the lack of well-defined basic notions was problematic. Before that, they were satisfied with primitive notions, but with the current level of sophistication it was not enough. Hilbert 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.

This is how we came up with ZFC: 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 Z, others prefer KFC, some guy prefers NBG, or KP. Oh, and what about New Foundations? It's a mess, but we can say that the majority of people consider ZFC=mathematics.

But then Gödel (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!

So this is how one can say that something is unprovable in ordinary mathematics: it just cannot be proved from those nine sentences.

And here come the large cardinals! What if we add to the nine sentences a tenth sentence, like, there exists an inaccessible cardinal. Then we can prove more stuff! 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!

Has it ever happened? It happens every day. The most famous (but controversial) case is Fermat's Last Theorem. 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 Grothendieck Universe, 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 is provable. (This is granted to start a flame war).

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, ineffable cardinals), so I'll end with a (partial) list of them completely without context for your amusement, from the weakest to the strongest:

Zero pistol

Icarus sets

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.

And now you can try this at home too! (*) Enjoy your time solving unprovable theorems with large cardinals!

(*) Keep out of reach of children. Keep away from food, drink and animal foodstuffs.
When using do not smoke. Avoid contact with eyes.  If swallowed, seek medical advice immediately and show this container or label.

You want to know what is the use of large cardinals? How about... proving God? Or just waisting a lot of time for an addicting game, that is important too.

Monday, September 8, 2014

Artemis Fowl and the large cardinals paradox (Part one)

Let's go back to young adults literature.

This time I am picking a sentence from the book Artemis Fowl: the Time Paradox.

It's the sixth book in the series Artemis Fowl, by the talented Eoin Colfer. 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, he said), 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.

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:

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.

Yes! YES! Infinitytimesinfinity times yes! This is so exact it hurts. Oh boy. I am not talking about the decades-long row on whether the Unfinished Symphony 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!

Well, be prepared, now, after reading this you will be one of those!

Pictured: not what I am talking about

Cardinals are just math-speech for numbers and infinities. In practice, they are the measure of the largeness of a set: if a set is finite it will be a number (finite cardinals), if it is infinite it will be some infinity (infinite cardinal).

Wait, if this is the first time you read this blog, then maybe you are not aware of Cantor's Theorem. Read this, 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.

But then what can be considered large for the people that are used to handle infinities like it's nothing? Here you can find some infinite sets that needs some serious brainpower to be visualized, and yet they are all extremely small. We have to pick an inaccessible cardinal, the smallest of the large cardinals, to just start to realize how large these guys are.

Again, not what I am talking about. Stop it.

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 as many as the cardinal itself. 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.

It is so big, that it trascends mathematics. With that I mean that it is completely outside mathematics, it cannot be reached with proofs. One cannot prove that it exists. 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, he is trying to prove theorems it is impossible to prove. How come?

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!

Cliffhanger!

(*) 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.

Aaaand, here is the second part

Monday, August 25, 2014

Remember this post? The one about infinity times infinity? The kindergarten kids? The infinities of dots? The pickets? That one.

Well, I have updates on that.

UPDATE 1:

Very cool gif, Google+! Thank you! This explains my thoughts much, much better. I don't know how you came up with this.

Wait.

How did you come up with this? 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 all 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, person!

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 gifs? What are you doing to my life, Google+?!

UPDATE 2:

It turns out that infinity times infinity is a thing, and there is a symbol for it.

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 (lemniscate) on top of the other, so it is more like infinity plus infinity. I imagine infinity times infinity as something more complex, a bit more mindblowing, something like this?

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, somebody invent a better symbol! Infinity times infinity deserves it!

UPDATE 3

This.

Did anybody said tattoo? Also, if you spend time on Google+, probably you have a lot of time for this.

Monday, August 18, 2014

These 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:

This is "Countably Infinite" by the A.G.Trio, 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 every song ever written, so nothing to see here), the key sentence that gives the title of the song is this:

There are countably infinite things I'd like to say to you

In this cases I think: what does a person that doesn't know what "countably infinte" means feels hearing this? Which images come to the mind? An infinite... you can count? Because in fact "countably infinite" is a pretty specific term in set theory.

Something is countably infinite if it is infinite and as big as the natural numbers, or aleph 0. That is, if you can connect every object to one and only one number, the objects are countably infinite. In this way, you can count the objects: object 1, object 2, object 3, ...  and this is way is called "countably". One way to do this is a list: if you can write the objects, one every line, without ever stopping, than you have countably infinite.

So, is it possible for the singer to have countably infinite things to say? Let's try to imagine the possible things he has to say as a list:

1. I love you
3. Don't leave
2. I'll miss you
3. My, that is a BIG pimple
4. It's like... purple
5. Hey, did you know that there is no word that rhymes with purple?
6. The sky is blue
7. I kissed a girl, and I liked iiit (of course)
8. Some infinities are bigger than other infinities...
9. BAM! Romantic
10. There are more stars in the universe than grains of sand on all the beaches of Earth
11. I love dinosaur erotica
12. No, no, no this is not related to the first sentence, don't leave!
13. Where am I?
14. I am sitting in a room, different from the one you are in now
15. But seriously, how would you say your skin is scaly from 0 to 10?
16. ...

Wait, this is taking too long. Let's do this more systematically. How many thoughts there are, at all? Better: let's count how many sentences one can write. 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 every sentence is associated one and one only number, and to every number one and one only sentence. For example: the number 1234567890 is the sentence "ceaantr" (almost Cantor!), while "I_love_you" is the number 76387629278892. So there are countably infinite possible things to say, one for each number!

But does the singer have really infinite things to say? Our brain is pretty big, but not infinite, it has somewhat around 2.5 petabytes (or a million gigabytes). So the thoughts he has in mind have to be finite! Come on, A.G. Trio! How could you even think that one guy had infinite thoughts! It's like you are purposedly exaggerating for...

Wait a minute.

That's it, isn't it? They didn't really 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?

...

Blimey! I always fall for this! I... I need time to think.

(Anyway, Cantor is 43868727)

Countable infinity is probaby much, much bigger than what you have in mind. Still, other things are even bigger.

Monday, August 11, 2014

Advanced Thinking & Thinking: Infinity times Infinity

Warning: the procedures in this post, if followed correctly, can lead to vertigo, psychological intoxication, and air-headedness. In other words: awesomeness.

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 paradise that Cantor provided us. I am going to describe some really simple infinite set, and the only thing you have to do is to try to imagine it. 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.

And have the right attitude. Which attitude, you say? This one:

This advertisement is incredible. It manages to introduce effortlessy a nice amount of infinite sets. But can you really image infinity times infinity? Try it now. I'll wait.

...

What took you so much? So let's see if my description is the same as yours.

This is one.

This is two (yeah, I know, stay with me).

And so on. Just adding one on the right, you are making +1.

Once you finished the numbers, you have the first infinite set! Congratulations! We call it ω. (Infinity! like the girl said).

Now, you know how to do ω+1: after ω, add one point.

Well, there is not much space there, let's go to a new line.

And now add another one (ω+2).

And so on, until you finished the numbers again. We have ω+ω! (Infinity plus infinity, like the guy said).

Now add another point, and continue. We have ω+ω+ω, i.e. three times infinity.

Go on, with four times infinity, five times infinity, and so on. At the end we have ωxω (Infinity times infinity! like the other girl said).

Great, this was not that difficult. So let's go deeper. I want you to imagine ωω, i.e. Infinity to the power of infinity! Let's do it visually.

Visualize again ω.

Now, between each two points, add ω. We have ωxω, i.e. ω2.

Do it again: between any two points in ω2, add ω. We have ω3.

Of course I am not going to make a drawing for that, don't be silly. 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 ω for each point in ω3), and again (ω4), and again! In the end you have in front of you ωω!

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...

When you counted them all, one picket two meters high appears. Then again the short ones, start counting again: one, two, three...

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!

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 ωω pickets.

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 ωω numbers.

Homework! How to visualize Infinity to the power of infinity to the power of infinity? And Infinity to the power of infinity to the power of infinity to the power of infinity? And Infinity made to the power of infinity infinite times? Have fun!

One personal note: it is disarming how practically all the children manage to think about infinity plus one, like above pretty much. 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! 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!

I think that the people that make the most fascinating jobs have still inside them the kid that they were. 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, si parva licet, I am still playing the game of "who says the biggest number".

And I am winning.

The story does not end here. You can find an update at this page. Also, did you know that there is a dance song about all of that?