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.
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.