[dwohnitmok]

This is subtle and a simple counting argument (definable means satisfies a finite formula, there are only countably many finite formulas, there are uncountably many reals, therefore there must be undefinable reals) doesn't work, because "definable in ZFC" is not something that is formalizable in ZFC and so the usual set-theoretic counting arguments don't work.

So it is in fact possible and consistent with ZFC that all reals are definable.

See: https://mathoverflow.net/questions/44102/is-the-analysis-as-...

[bubblyworld]

Thanks, that's a wonderful link and a nice puzzle to think about. The best intuition I have for it is that since the predicate "isDefinableReal(x)" is not itself definable in first-order set theory, there is no way to construct the set of all definable reals in the first place. Thus saying it's countable is basically meaningless - what, exactly, is countable?

[drdeca]

If you use ZFC+Consistent(ZFC) as your meta-theory, and within it consider a model of ZFC, then surely one can consider the set (in the meta theory) of sentences which pick out a unique real number in the model, and then the set of real numbers in the model which are picked out by some sentence? It might not be a set that belongs to the model, but it’s a set in the meta-theory, right?

And, I imagine that the set of real numbers of the meta theory could be (in the meta theory) the same set as the set of real numbers in the model?

[bubblyworld]

You can do this, but things get strange in the meta-theory. Some models of ZFC are countable according to the meta-theory! And some of them have models of the reals that are countable according to the meta-theory. There's no contradiction here, because what the meta-theory thinks "countable" means has nothing to do with what the inner model thinks "countable" means.

(for an extreme example of this, by the Löwenheim–Skolem theorem there are countable models of ZFC)

So you can do what you are suggesting, and you will of course get a countable set of reals (or what are reals according to the inner model), but they might not be countable according to the inner model. They might not even be a set according to the inner model, and there are even inner models that think you've got all of the reals!

(see https://mathoverflow.net/questions/351659/set-of-definable-r... pretty heavy reading)

So the statement "the set of definable reals is countable" is nonsense - you're talking about things that live in different universes of meaning.

[drdeca]

I think there should be models of ZFC in which the set of reals of the model is, in the meta-theory, the same object as the set of reals of the meta-theory.

And I think by virtue of this, the statement should have meaning.

As like, a statement in the meta-language that models of ZFC which have as their sets of reals, the (according to the meta-theory) set of reals, that the set of reals definable within ZFC, is a countable set of the meta-theory.

Also, did someone downvote your comment?? I don’t know why if so. It seems a productive comment to me.

[bubblyworld]

They can have the same set of reals (by construction, for instance) but they won't behave the same way as sets (the membership relation will be different). I think you need to be very clear about what you are doing here.

By definition an inner model consists of some domain (a set of sets) and some choice of mappings from all the function/relation symbols of ZFC to functions/relations on this domain, satisfying the axioms of ZFC.

You are suggesting to enumerate every formula of ZFC, evaluate them against this inner model, and take the set of all reals that are uniquely picked out by some formula (according to the model).

The trouble is that even though you can make the set of reals the same, your chosen interpretation of all the functions/relations will not match the meta-theory, and in fact cannot match it (i.e. the meta-theory cannot provably construct an inner model like this, by Tarski's undefinability of truth theorem).

So you will get a set of reals, and they will be reals according to the meta-theory too, but the meta-theory cannot relate this set to the definable reals of the meta-theory.

As far as I can see this is the strongest statement you can actually prove: "the set of reals in any inner model of ZFC uniquely definable by a formula (according to the interpretation of the inner model) is countable (according to the interpretation of the meta-theory)".

> Also, did someone downvote your comment??

Someone did, yeah, but I don't mind =) I probably sound like a crackpot to the uninitiated.