Transitive models and CH

Question

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.

Answer 1

This is possible. Let me first give a close result by Philipp Schlicht and myself, and then give the relevant result by Woodin.

Let $V$ be the first Cohen model. Namely, add $\omega$ Cohen reals to $L$ and take the finitary permutations of $\omega$ applied to "switching" the reals, and all the sets definable from finitely many reals and the set of Cohen reals. Then $A$ is the set of Cohen reals is Dedekind-finite.

If $\kappa$ is any infinite $\aleph$ cardinal, then forcing using finite injections from $A$ to $\kappa$ recovers choice and in fact gives us that $A$ is the generic filter for adding $\kappa$ Cohen reals over $L$. In particular no cofinalities were changed.

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):

For any regular uncountable $\kappa$, $L(\Bbb R)^{L^{\operatorname{Add}(\omega,\kappa)}}$ is the same. In particular, if you start with $L$, add $\omega_1$ Cohen reals, consider $L(\Bbb R)$ [which is a model of $\sf DC$], you can recover choice without adding reals by ordering $\Bbb R$ in however way you like it to be ordered.

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.