How do I explicate A is countable ?

[mpitluk]

How do I explicate "A is countable"?

My attempt:

In set theory, every thing is a set, even functions. Thus when we say "A is countable in M" we mean that there is another set B in M that contains {naturals} and A as ordered pairs.

I'm having trouble spelling out the "as ordered pairs" part.

Is is: B in M that contains N and A in ordered pairs such that (a, n) where a ∈ A and n ∈ N and for every a there is exactly one corresponding n and for every n there is exactly one corresponding a?

Is there an easier way to write this?

 

[Vargo]

When you say "A is countable in M", you mean A is a countable subset of M, right?

Okay then that means that there is a bijection between A and N. So there exists a set of ordered pairs {(a,n) | a belongs to A and n belongs to N}. For each a in A there is exactly one corresponding ordered pair whose first component is a (this makes it a function), and likewise for each n in N (which makes it bijective).

This set corresponding to the bijection is itself a subset of the Cartesian product M X N.

If you want to break this down one step further then (a,n) = { {a},{a,n}}. So the ordered pairs are technically subsets of the power set of (M union N).

 

[mpitluk]

When you say "A is countable in M", you mean A is a countable subset of M, right?

Sorry, I wasn't being clear. I meant that M is a model that satisfies the sentence "A is countable." But, I don't think that affects anything else you wrote as models are sets as well. Thanks!