## 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.