Oh. Thanks.
Are there any examples of different models of ZFC that contain the same reals?
Well, models can have the same reals by fiat. If I cut off an existing model below an inaccessible, I certainly haven't changed the reals. Alternately I could restrict to the constructible closure of the reals L(R), which satisfies ZF but generally fails Choice (you don't expect to have a well-ordering of the reals in this model).
I think, though, that Stuart_Armstrong's statement
Often, different models of set theory will have the same model of the reals inside them
is mistaken, or at least misguided. Models of set theory and their corresponding sets of ...
With thanks to Paul Christiano
My previous post left one important issue unresolved. Second order logic needed to make use of set theory in order to work its magic, pin down a single copy of the reals and natural numbers, and so on. But set theory is a first order theory, with all the problems that this brings - multiple models, of uncontrollable sizes. How can these two facts be reconciled?
Quite simply, it turns out: for any given model of set theory, the uniqueness proofs still work. Hence the proper statement is:
Often, different models of set theory will have the same model of the reals inside them; but not always. Countable models of set theory, for instance, will have a countable model of the reals. So models of the reals can be divided into three categories:
And similarly for the natural numbers.