Recent content by mpitluk

How is uncountability characterized in second order logic?

Thanks for this great answer. Why is it impossible for a standard model to say of itself that it is countable?

Can someone point me to a first-order axiomatization of ZFC? As I've mostly seen ZFC expressed in higher-order logics.

"The paradox: Let T be a standard first-order formulation of ZFC. Assume T has a model. By Skolem's Theorem, T has a countable model M. Since T ⊢ ∃A(A is uncountable), M ⊨ ∃A(A is uncountable). But how can M—i.e. a model that “sees” only countably many things in the universe—“say” some sets...

Must all models of ZFC (in a standard formulation) be at least countable? Why I think this: there are countably many instances of Replacement, and so, if a model is to satisfy Replacement, it must have at least countably many satisfactions of it. Does my question only apply to first-order...

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!

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"...

Whoops. Right, thanks.

So Infinity, Empty-Set and Pairing are jointly sufficient and individually necessary for P(N)?

Is it true that for every standard formulation T of ZFC, T ⊢ the power set of {naturals}? After all, the empty set axiom and the pairing axiom are in T, and so we get N. Then by the power set axiom we get P(N).

What is a standard model?--a model that satisfies the standard first-order ZFC axioms? And what is an infinite decreasing series of sets? Also, must a countable model contain the natural numbers? It seems like it is not necessary, as we have been saying that (typically?) saying a model is...

Can a model be countable from its own "perspective"? I'm reading about Skolem. And I'm wondering about the result of the paradox: that countability (at least in first-order formulations) is relative. Now, even when we state Skolem's theory -- if a first-order theory has an infinite model then...

Sweet, I didn't know that either. Nice. I didn't get it at all at first, as I had no idea what mod arithmetic was. But I got it (both, really) shortly after. So, if we are "working" mod 4, 2 + 3 = 1? So, this equivalence is with respect to something right? They act as if they have the...

I didn't know that a cardinal number is an equivalence class of ordinals. I'm not entirely sure what this means. For example, you have some cardinal number, say, aleph_1 and then you have some set or class of all countable ordinals. Now, all I know is that the set of all countable ordinals...

I'm totally new to ordinal numbers, I'll have to check out that book. It's seems you're saying the following. An ordinal number corresponds with a certain cardinality and these correspond with the number classes. For example, the set of all finite ordinals (each corresponding to the different...