ZFC + not-CH, can a set with cardinality between $\aleph_0$ and $2^{\aleph_0}$ be defined?

Continuum hypothesis states, there is no set with cardinality between the integers and the reals.

There is a milestone result, that CH is independent from ZFC. That means, both of ZFC + CH, and ZFC + not-CH are consistent.

What if ZFC and not-CH. Thus, we have an axiom which states, there is a cardinality between $\aleph_0$ and $2^{\aleph_0}$.

Can a such set be defined?

Solutions Collecting From Web of "ZFC + not-CH, can a set with cardinality between $\aleph_0$ and $2^{\aleph_0}$ be defined?"

In some sense, yes, you can always construct a set of size $\aleph_1$. Specifically $\omega_1$ is a set of size $\aleph_1$. And if the continuum hypothesis fails, it serves as a counterexample.

You might want to ask whether or not you can construct a set of real numbers of this particular size, and the answer to that will depend on your notion of “construct”, but if you mean define “in a reasonable way” the answer is consistently negative.