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Ralsina.Me — Roberto Alsina's website

Infinite Amateurism on Maths

I stum­bled up­on Thomas Thur­man's post on his blog (here) where he com­ments about a dis­cus­sion on The Guardian about how .99(re­cur­ring) is or is not the same as 1.

Of course to any­one who knows what a ra­tio­nal num­ber is, .99(re­cur­ring) is sim­ply a very long way to write 1. Hel­l, to any­one who both­ered learn­ing his frac­tion­s, that should be ob­vi­ous!

But any­way, one of the com­ments men­tions Hilbert's Hotel, which is a pet toy of mine.

If you are un­com­fort­able or an­noyed by the con­cept of in­fin­i­ty, you may want to avoid the rest of this post.

Hilbert's Ho­tel is this para­dox (Thanks Wikipedia!):

A ho­tel with an in­fi­nite num­ber of rooms (1, 2, 3 and so on, so it's a nu­mer­able in­fin­i­ty) is ful­l. A guest ar­rives. Yet he still gets a room. How?

The an­swer is that you ask the guest in room 1 to move to room 2, from 2 to 3 and so on. Then the new guest goes to room 1, which is now free.

Be­cause of the na­ture of in­fin­i­ty, this work­s, while on a fi­nite ho­tel it would­n't.

  • Un­in­­tu­i­tive things about in­­fin­i­­ty: If you add any num­ber to it, the re­­sult is the same in­­fin­i­­ty.

Now as­sume an in­fi­nite (nu­mer­able) num­ber of guests ar­rives. You have to ask the guest in room 1 to go to room 2, the guest in room 2 to go to room 4, and guest n to go to n*2.

Now you have an in­fi­nite (nu­mer­able) num­ber of free room­s: all the odd room­s.

  • Un­in­­tu­i­tive things about in­­fin­i­­ty: If you mul­ti­­ply it by any num­ber, the re­­sult is the same in­­fin­i­­ty.

  • Un­in­­tu­i­tive things about in­­fin­i­­ty: If some­thing is in­­finite, it con­­tains a part the same size as the whole (fol­lows from the pre­vi­ous two things, and is, in fact an "if and on­­ly if").

How­ev­er, here's where it gets trick­y. You could get a cer­tain num­ber of new guests and there could be no way to fit them in the rooms even if the ho­tel was emp­ty.

Be­cause there's in­fin­i­ty, and then there is in­fin­i­ty. You saw that when­ev­er I men­tioned the num­ber of rooms I men­tioned they were in­fi­nite (nu­mer­able)? That's be­cause you can put an in­te­ger num­ber to each, and num­ber them al­l.

There are in­fi­nite sets of things that are big­ger, they are lit­er­al­ly un­count­able. You can't put a num­ber to each, even with an in­fi­nite amount of time (and yes, I know that in­fi­nite amount of time there is a big prob­lem).

The sim­plest set imag­in­able that large is that of the re­al num­ber­s. The re­al num­bers are all the num­bers you can imag­ine, al­low­ing for in­fi­nite dec­i­mal­s, and al­low­ing that those dec­i­mals may not ev­er be re­cur­ring (so you have things like 2, 1/3, and pi).

Show­ing there are more of those that there are in­te­ger num­bers is not sim­ple enough for this but go along with me for a while.

  • Un­in­­tu­i­tive things about in­­fin­i­­ty: There are dif­fer­­ent sizes of in­­­fi­nite. Go blame Georg Can­­tor.

Now it gets re­al­ly weird. Sup­pose we call the size of the in­fi­nite in Hilbert's Ho­tel A0 (I have no idea how to do an Ale­ph, sor­ry), and the size of the re­al num­bers C. Can­tor showed how to build, once you have an in­fi­nite set, a larg­er in­fi­nite set called his pow­er set.

That means we now have a whole in­fi­nite (nu­mer­able) "sizes" of in­ifi­nite things. Those are called the trans­fi­nite num­ber­s.

  • Un­in­­tu­i­tive things about in­­fin­i­­ty: There are in­­­fi­nite dif­fer­­ent sizes of in­­­fi­nite. Go blame Georg Can­­tor some more.

Which brings a lot of ques­tion­s:

  • Are there on­­ly those? Is­n't there some­thing be­tween A0 and C which is some in­­-­be­tween size?

  • Is there an in­­­fi­nite set that's smal­l­­er than the in­­te­ger­s?

  • Ok, in­­­fi­nite in­­finites... in­­­fi­nite (nu­mer­able) in­­finites, or in­­­fi­nite (some­thing else) in­­finites?

Well... I have no idea. And last I checked, which was long ago, and my mem­o­ry is no good, noone else knew.

This is the kind of things that will tell you whether you could be a math­e­ma­ti­cian. Do you find all this talk about trans­fi­nite num­bers in­trigu­ing and mys­te­ri­ous, or just dull and bor­ing and im­prac­ti­cal?

If you find it dull and bor­ing, it may be my writ­ing, or you may be un­suit­ed for math­s.

If you find it in­trigu­ing and/or mys­te­ri­ous, it cer­tain­ly is not my writ­ing, and you would prob­a­bly en­joy maths in your life. Where else are you go­ing to run in­to "noone knows that" this quick­ly?

The prob­lem is, of course, that most of the fun math has al­ready been done at least a cen­tu­ry ago, but there is al­ways a chance of some­thing fun and in­trigu­ing and new com­ing along.

The last I know of was Gödel's the­o­rem, which is re­al­ly sim­ple enough for any­one with knowl­edge of arith­metic to fol­low, but weird enough for 99.99% of the peo­ple to go crazy about (and for those who don't re­al­ly un­der­stand it to write whole books about it ap­ply­ing it to to­tal­ly im­prop­er sub­ject­s).

But you know, noone re­al­ly had thought of such thing as "larg­er than in­fin­i­ty" quite as Can­tor did, and noone thought about Gödel's sub­ject quite as he did be­fore him.

Maybe we are miss­ing some­thing ab­so­lute­ly sim­ple, in­cred­i­bly el­e­gan­t, awe­some­ly shock­ing some­where in ba­sic math­s. Not like­ly. But pos­si­ble. Would­n't it be fun to find it?

BTW: Gödel starved him­self to death and Can­tor "suf­fered pover­ty, hunger and died in a sana­to­ri­um".

Roflech / 2006-06-22 17:30:

Hi, I really couldn't get pass the hotel thing.

Obviously I´m not suited for maths, which can be a explanation why I became a lawyer. Or maybe it´s just because I'm a woman. I read once that women are less suited for maths than men, because they tend to be more practical, while men are better with abstract ideas. That sounds quite sexist, but It proves right on me, at least.

But what I can surely say is that I wouldn´t spend two minutes in a hotel that pretended to be changing my room indefinitely. So the whole idea is pointless or it´s just a bad example. Where is the person who thought about it pretending to get an infinite bunch of nothing-to-do-with-their-time people to try such an experiment?

I know that this is just theoretical, nothing pretended to be done really, and that´s is mostly what Í don't understand about maths. What's the use of thinking about things that have no practical use???
No wonder those mathematicians starved to death, no matter how simple, elegant or shocking their findings where...

Anyhow I guess maths must be important in some kind of way. I'm just glad to leave the task to others.

Eimai / 2006-06-22 17:33:

"Isn't there something between A0 and C which is some in-between size?"

well, the reason nobody knows is because it is proven that it can't be proved. In the usual mathematical axiom system (ZFC - zermelo-fraenkel and axiom of choice) the question

2^(aleph0) = aleph1 ?

(with aleph1 the first set bigger than aleph0 in ZF, and 2^(aleph0) the number of subsets in aleph0, which is the number of real numbers i.e. your C)

is equivalent to the continuum hypothesis, meaning that it cannot be proven in ZFC. Either you accept it, or you don't.

It's like the axiom of choice: you can't prove nor disprove it with ZF (only zermelo-fraenkel), and some mathematicians don't accept the axiom of choice and develop their own math system, the same with the continuum hypothesis.

"Is there an infinite set that's smaller than the integers?"

assuming you work in ZFC, no there isn't.

Anonymous / 2006-06-22 19:18:

@Roflech - You said "What's the use of thinking about things that have no practical use?"

If you think that way, all art is useless. Whats the practical use of having Monalisa?

Feynman once said about Physics (same applies to Math): Physics is like sex. Both may produce practical results, but that's not why we do it.

Roberto Alsina / 2006-06-22 23:11:

@Roflech: I could tell you why maths are important, but I have already tried. It works about as well as you explaining me politics ;-)

@Eimai: Nice to see someone who actually knows the subject around ;-)

So, within ZFC, we know some things, and not everyone accepts ZFC. Sounds a lot like "we have no idea" to me, for some value of "we" (and some value of idea ;-).

Anyway: I am still waiting, maybe someone can figure out an axiomatic system as simple as ZFC where all this makes more sense.

And anyway "the continuum hypothesis" is one of my top 5 math thing names. It's so startrekky.

Marc / 2006-06-23 01:01:

There are other set theories besides ZFC. If you want one where the continuum hypothesis is true, you can just add it on as an axiom, and then ask deeper questions -- does there exist yet another infinty between C and the newly added one? (Yes, and all the infinities you could ever want.)

You can also work in another mathematical universe -- a different topos -- where things can be wildly different.

@Alsina: It's not so much that some mathematician's don't accept ZFC, it's just that sometimes it makes sense to work with or without some axiom, such as the axiom of choice. In this case, since choice is independent of ZF, we are free to accept or reject it whenever necessary. ZFC is logically consistent, so it is not rejected on a basis of being incorrect.

@Roflech: As far as why these things are important -- you need uncountable infinities to make calculus work, and last time I checked calculus was being used to make just about everything.

And for infinities of infinities of infinities... 2^infinity is always a bigger infinity, and you can take infinities^infinities, and infinite number of times if you want, and it turns out that there is no largest infinity. See large cardinals on Wikipedia.

Arjun / 2006-06-23 02:47:

Well, another reason some mathematicians do not include the axiom of choice in their reasoning is that where it is not necessary, a proof is strengthened by its absence - that is, it requires a smaller set of axioms (ZF) to prove whatever is at hand.