The predicate “is a real number” is absolute for transitive models of ZFC in the sense that if M and N are such models with M contained in N, then for every element x of M, the two models agree on whether x is a real number. But it can certainly happen than N has more real numbers than M; they just have to lie completely outside of M.
Example 1: If M is countable with respect to N, then obviously M doesn’t contain all of N’s reals.
Example 2 (perhaps more relevant to what you asked): Under mild large cardinal assumptions (existence of a measurable cardinal is sufficient), there exists a real number 0# (zero-sharp) which encodes the shortcomings of Gödel’s Constructible Universe L. In particular 0# lies outside of L, so L does not contain all the reals.
Thus if you started with L and insisted on adding a measurable cardinal on top, you would have to also add more reals as well.
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 reals are extremely pliable, especially by the method of forcing (Cohen proved CH can consistently fail by just cramming tons of reals into an existing model without changing the ordinal values of that model’s Alephs), and I think it’s naive to hope for anything like One True Real Line.
The predicate “is a real number” is absolute for transitive models of ZFC in the sense that if M and N are such models with M contained in N, then for every element x of M, the two models agree on whether x is a real number. But it can certainly happen than N has more real numbers than M; they just have to lie completely outside of M.
Example 1: If M is countable with respect to N, then obviously M doesn’t contain all of N’s reals.
Example 2 (perhaps more relevant to what you asked): Under mild large cardinal assumptions (existence of a measurable cardinal is sufficient), there exists a real number 0# (zero-sharp) which encodes the shortcomings of Gödel’s Constructible Universe L. In particular 0# lies outside of L, so L does not contain all the reals.
Thus if you started with L and insisted on adding a measurable cardinal on top, you would have to also add more reals as well.
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
is mistaken, or at least misguided. Models of set theory and their corresponding sets of reals are extremely pliable, especially by the method of forcing (Cohen proved CH can consistently fail by just cramming tons of reals into an existing model without changing the ordinal values of that model’s Alephs), and I think it’s naive to hope for anything like One True Real Line.
Thanks for that elucidation.
Thank you for helping me fill my stupid gap in understanding!