Comments on: Recognising countable sets

By: elacaraq

elacaraq — Tue, 26 Aug 2008 21:02:33 +0000

Dear Gowers: Yes, I realized the problem in my polynomial example. What I had in mind was to map each polynomial to its degree. So for each n there are countablely many polynomials with degree n. However this needs proof too. I see that your weaker statement is actually more convincible.

By: gowers

gowers — Tue, 26 Aug 2008 08:26:11 +0000

elacaraq, the stronger statement that A is countable if there’s a function to such that every n has countably many preimages is true too, and is another way of saying that a countable union of countable sets is countable. But I prefer to tie my hands behind my back and use just the weaker statement because I like the proofs that result.

In the polynomials case, if you map each polynomial to the coefficient of its highest term, then it’s not all that easy to prove that the preimage of n is countable. Indeed, given any polynomial P of degree d, you can add the polynomial to it and obtain a polynomial with leading term n, so proving that the preimage of n is countable is precisely as hard as proving that the set of all polynomials with integer coefficients is countable, which is of course the problem we started with.

By: elacaraq

elacaraq — Tue, 26 Aug 2008 06:08:40 +0000

“If you can find a function from A to such that every n has finitely many preimages, then A is countable.”

Can we just say, if we can find a function from A to such that every n has countably many preimages, then A is countable? This way A is indeed a countable union of countable sets.

Like the polynomial example, if we map each polynomial to the coefficient of its highest term, then every n have as its preimages, hence the set of all polynomials is countable.

By: A small countability question « Gowers’s Weblog

A small countability question « Gowers’s Weblog — Sun, 10 Aug 2008 15:12:18 +0000

[...] davidspeyer pointed out in an interesting comment on the previous countability post, there do exist explicitly definable sets that we can show are countable even though we cannot give [...]

By: Per Vognsen

Per Vognsen — Wed, 06 Aug 2008 08:36:50 +0000

“If you can find a function from A to \mathbb{N} such that every n has finitely many preimages, then A is countable.”

It seems beneficial to generalize this ever so slightly to say, if you can find a function from A to some countable set B (of which N is the prototype and exemplar) such that every x in B has finitely many preimages, then A is countable.

To demonstrate:

“To prove that the set of all polynomials with integer coefficients is countable is a similar exercise, but slightly more complicated.”

In your proof, you are essentially composing together three smaller, simpler proofs and three individual functions:

f : Z[x] -> N[x]

This takes absolute values coefficient by coefficient. A polynomial of degree n in N[x] can have no more than 2^(n+1) preimages, for there is at most a two-choice sign ambiguity for each of the n+1 coefficients.

g : N[x] -> NxN

This takes a_0 + a_1 x + … + a_n x^n to (n, a_0 + a_1 + … + a_n). Actually, a simpler way would be to take the image as (n, max {a_0, a_1, …, a_n}), for then (x,y) is easily seen to have no more than (y+1)^x preimages.

h : NxN -> N

This takes (x,y) to x+y. This is the function you already used in your proof of NxN’s countability.

By: davidspeyer

davidspeyer — Fri, 01 Aug 2008 15:21:04 +0000

Regarding the claim “I cannot think of any explicitly defined countable set that isn’t completely obviously countable by this criterion.”: the set of zeros of the Riemann zeta function.

This is countable by another standard trick: to prove that a subset of (or ) is countable, show that it has no accumulation point. Indeed, the zeroes of any nonzero complex analytic function are countable for this reason.

By: Mark Meckes

Mark Meckes — Wed, 30 Jul 2008 17:05:55 +0000

Tim, that sounds even better.

By: gowers

gowers — Wed, 30 Jul 2008 15:47:19 +0000

Mark, looking at what I wrote again, I find that I completely agree with you. But I think that on the site itself the layout will be different anyway. Probably there will be a little column on the side with information about things like prerequisites, the intended audience, the area(s) of mathematics to which the article is relevant, keywords, and so on.

By: Mark Meckes

Mark Meckes — Wed, 30 Jul 2008 14:46:41 +0000

I haven’t looked carefully at the content yet, but I think the basic prerequisites-quick description-general discussion-examples layout is great. I think it might be even better, though, to put the quick description before the prerequisites. That would seem to me to better reflect the way we often learn and use new tricks in math: first you figure out what you’d like to be able to do, and then you fill in whatever background you’re missing to understand how to do it.

This suggestion may seem somewhat silly with regard to the sample articles above, given that you need the definition of countability to appreciate the titles themselves, but there will probably be many tricks that involve ideas that don’t appear explicitly in the title or even in the quick description.