Suppose $M, N$ are two countable transitive models of ZFC which have same ordinals, cofinalities and reals (but not necessarily same sets of reals!). Suppose $M$ models the continuum hypothesis. Can we conclude that $N$ also models continuum hypothesis? I can see that if this fails, then neither one of $M, N$ is included in the other. But what if $M, N$ are incomparable.
2025-01-13 09:01:31.1736758891
Transitive models and CH
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This is possible. Let me first give a close result by Philipp Schlicht and myself, and then give the relevant result by Woodin.
This is from a work in slow-progress. But it shows that you can have more Cohen reals by forgetting how to count them first, and then trying to count them in a different way. Our result, however, is not what you are asking about, because once you move from Dedekind-finite set of reals to a Dedekind-infinite set of reals you invariably add new reals.
In a conversation with Woodin about a month ago, I mentioned this result, and he told me that some 30 years ago he proved the following (and never published it):
This means that you have a plethora of various models with the same ordinals, cofinalities, and reals, but in different models you have different reals. Which is what you wanted.