Comment by bubblyworld
6 months ago
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?
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?
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.
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.
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