## Background

In constructive set theory (say based on CZF) there are inequivalent ways of stating the continuum hypothesis. Some of them are easily if not trivially refutable with common anti-classical assumptions. For instance:

Theorem: The class of all subsets of $\mathbb{N}$ is not in bijection with the class of all hereditarily transitive countable sets. (Clearly classically equivalent to ~CH)

Proof (CZF+REA+SC): The class of all hereditary transitive countable sets is a set. But the class of all subsets of N is a proper class (because SC: all sets are subcountable -- the anti-classical assumption). QED

But the same subcountability argument shows the class of all subsets of $\mathbb{N}$ is not in bijection with the set $2^\mathbb{N}$, nor with the set of real numbers. And anything that implies UZ (continuum is undecomposable) will imply that the set of real numbers is not in bijection with $2^\mathbb{N}$ either. So there are some classically equivalent but constructively inequivalent natural variants on CH there.

And of course CH can be rephrased without $\aleph_1$. We can ask whether there is a set which embeds into the continuum and into which the integers can be embeded, but neither converse holds. Constructively this is a weaker hypothesis, giving another "dimension" of natural variants.

It seems plausible to me that all of these variants are false under SC or UZ or intuitionistic continuity principle, or common assumption like that.

## Question

Does anyone know of any simple statements that are classically equivalent but constructively inequivalent to CH, and which are true, or open problems, under common anti-classical assumptions like subcountability or unzerlegbarkeit?