Some infinities don't exist more than other infinities (an ultrafinitist point of view)

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 this post.

Hahaha. Infinity.

What a joke.

Only fools 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!

Infinity is just an illusion. God is dead, 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, not so much. We thought the speed of light was infinite, not even close. We thought that between two points there were infinite points, or that between two instants there were infinite moments of time, wrong again and again. Everything around us is finite. You don't go to the pet shop and buy infinite kittens. That would be too cuteness for anybody.

Spoiler alert: this is not possible.

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!

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 exist. Just because you can imagine 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.

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.

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!

Enough with this infinite foolishness, then! Eat your vegetables, and embrace the world as it is! (Vegetables are delicious, too! Especially the Brussels Sprouts.)

Infinite Mess (Part Two)

This is the second part of a post that started here. 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.

So, what did DFW wrote(*)? Here is the source, verbatim:

The Continuum Hypothesis gets characterized in all kind of different ways: [...](**) Is the same as ? 

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:

Natural numbers 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.

And one picture

Real numbers 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.

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 Ein Beitrag zur Mannigfaltigkeitslehere, 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 Continuum Problem, 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 Continuum Hypothesis, 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.

So? Understood? If you read DFW's book I'd really like to know if it was easier to understand than this.

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?

 is the size of the set of the real numbers, i.e., how many real numbers there are.

is the size of the set of the sets of natural number (I'll stop you before you start Xzibit memes), i.e., how many sets of natural numbers there are.

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.

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? 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.

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...

This is mine. Don't judge me, the situation at the office is awkward enough.

and concentrate on the following (***).

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,

In this way

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!

Let's see first that the real numbers cannot be less than the sets of natural numbers. Pick a set of natural numbers

then draw it in the number line

write a 0 when it's empty, 1 when it's full

and finally add 0. at the front.

There you are, for each set of natural number, you can write a different real number. So real numbers cannot be less.

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:

Therefore the two things do have the same size, and the climax of DFW's booklet is an epic fail.

Really, it's disappointing. It started so well, with Zeno's paradox of the turtle and everything...

Wait a minute, where did I see the turtle paradox and Cantor work together in the wrong way? John Green is a big fan of David Foster Wallace, right? Maybe...

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!

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! That's where everything starts! That's a scoop!

What? What are you saying? You mean... he already admitted that in the FAQ page I have already linked once? 

Oh.

I'll see myself out.

(*) Since I am taking for granted that you, reader, are a DFW fan, I am adding lots of footnotes. Have fun!
(**) 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.
(***) Or don't. Really, you can just skip the whole paragraph, you, reader, are the king, because I am so post-modern.
(****) In the finite case, it means that the apples are less than the oranges, but infinity is weird.

Thanks to Gabriel for pointing this out to me.

Infinite 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.

The reactions of literary fans are known to be sober

I'm not going to be only critical, I am going to destroy one of his works (well, at least some lines). I am talking about David Foster Wallace, the brilliant mind behind Infinite Jest, an encyclopedic, metamodernist, hysterical realist 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.

I will use his words to defend myself:

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. 

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!

Unfortunately, it seems that Foster Wallace didn't go where I've been. In 2003 he wrote a booklet, Everything and More, 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?

Disclaimer: I haven't read it. What I read are the critical reviews of Rudy Rucker and Michael Harris, 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. There is always a misunderstanding when writers try to explain mathematical concepts: the literary way of dealing with concepts is through vagueness.  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 perfection of its concepts. It's a huge mechanism full of cogs and wheels, and yet everything works perfectly, every minimal part is on time. It's like juggling, or rock balancing: the slightest error can ruin everything.

Literature vs Mathematics
(Left: (c) Frank Grisdale)

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 unreliability of the narrator. And he admits that in this book! So, what's the point of reading a mathematical book where the writer always tries to trick you? 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.

Well, then, that is how it is written. But what about the content? Surely there is a lot to learn reading this book, right?

No.

Unfortunately, there are many, many errors. There are websites 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 had a direction, that all of this was to reach a particular point, You turn the page ready to read it, no, to experience it...

and it is wrong.

The musical equivalent

What? You don't expect me to explain the climax now, do you? Just go here to read about this crime against infinity.

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. 

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

Cada cuerpo tiene

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. 

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?

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.