A question regarding the status of CH in the Gitik model

Question

Consider models of ZF+"Every uncountable cardinal is singular" (eg. Moti Gitik: "All uncountable cardinals can be singular", Israel journal of Mathematics, 35(1-2): 61-88, 1980). How should CH be formulated in such models? Can CH be false in such models and if so, how many intermediate cardinalities between $\omega$ and $2^{\omega}$ can there possibly be since such models are choiceless? I apologize in advance for the ignorance of the question--it is asked honestly. (As regards how CH should be formulated in such models I may have answered my own question if in fact such models are indeed choiceless).

Answer 1

(This is a long comment at the moment.)

To my best of knowledge pretty much all these questions are somewhat open.

By somewhat I mean that we don't quite know the answer for Gitik's model, or for "most models", or many limitations on what can happen.

We do know that it is consistent that $\omega_1$ is singular and $\sf CH$ holds in the sense that there are no intermediate cardinalities between $\omega$ and the continuum (e.g. in Truss' model, where the perfect set property holds). But it can be that the continuum itself is the countable union of countable sets and the continuum hypothesis fails in the sense that there are intermediate cardinals (as shown by Miller to be true in the Feferman-Levy mode). It should be noted that if the continuum is the countable union of countable sets, then $\omega_1$ is singular.

To my knowledge, it's not even clear whether the real numbers are a countable union of countable sets in Gitik's original model, (see the remarks below) and finding how many cardinalities are between cardinals is often a daunting task.

Here is a related MathOverflow question.

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Some additional remarks.

1. Gitik actually proves in his paper (the last theorem) that the continuum is not a countable union of countable sets. He also proves that every set is the countable union of smaller sets. In particular the continuum is the countable union of sets which have cardinality smaller than the continuum, but are uncountable. Therefore the continuum hypothesis is false in Gitik's model.

   On the other hand, this does not tell us how many intermediate cardinals are there. It might be just one, or it might be many. This fact does not appear in the original paper, and I don't think it has a definitive answer elsewhere in the literature.

2. Miller proves in his paper about long Borel hierarchies (the online version, in the appendix) that if one modifies the collapse of $\aleph_\omega$ then one gets a model which is "similar to Gitik's".

   • This seems closer to what Truss did in his model, where the continuum hypothesis does in fact hold. But not quite.

   • It's still unclear whether or in this model, or in Gitik's, the continuum hypothesis does hold.