Are there any constructive axioms which disprove the continuum hypothesis?

Question

I understand that the Continuum hypothesis is independent of ZFC, so that we may comfortably add either the continuum hypothesis or its negation to ZFC without creating any paradoxes (unless ZFC had them to begin with), and in fact there are several large Cardinal axioms that are inconsistent with CH.

My question is this: are there any proposed additions to ZFC which not only imply the negation of CH, but in fact allow for the explicit construction of a set with cardinality between $\mathbb{N}$ and $\mathbb{R}$?

If not, could such axioms be lurking about? Do we have any idea what such sets may "look" like?

I know that there are quite a few questions on CH lying about on this site, so if I missed a duplicate of this question I'd appreciate being directed to it as much as an answer.

Answer 1

Carl's comments are exactly right. However, in a different sense there are indeed serious obstacles to finding "explicit" sets with intermediate cardinality:

Every analytic set $X\subseteq \mathbb{R}$ satisfies the perfect set property - that is, either is countable or contains a nonempty closed set with no isolated points (and hence has size continuum).

The analytic (=${\bf \Sigma^1_1}$) sets are those which are the continuous image of some Borel set. The situation gets worse if we assume large cardinals:

Assuming some large cardinals, every projective set of reals has the perfect set property.

The projective sets of reals are all those which can be described using quantification over reals. In practice, basically every set of reals you can think of is Borel - that is, much simpler than analytic.

If you are interested in results like these, you should look at Descriptive Set Theory.