Cardinals in $ZFC+\neg CH$ Before  asking my question I like to admit that i am not an expert in the foundation of mathematics and I am interested in this issue more from a philosophical perspective.  So my question may be competently naive. If we assume $ZFC$ and the negation of the continuum hypothesis, what is know about the set $$\{|A|~|~A\subseteq \mathbb{R},~|\mathbb{N}|<|A|<|\mathbb{R}|\}.$$ Is in finite, countable or uncountable? Or are these claims independent of $ZFC+\neg CH$. As a Platonist in the philosophy of mathematics I suppose that the set is uncountable. Everything that possibly exists, exists.  But what is the consistency strength of such a claim?

Thanks for the answers so far. Perhaps I should make my last question more precise. Is ZFC plus the assumption that the set above is uncountable consistent if ZFC is?  
 A: By a theorem of Solovay, $|\mathbb R|$ can consistently be $\aleph_\alpha$ for any ordinal number $\alpha>0$ that does not have countable cofinality. Then the set $\{|A|:A\subseteq\mathbb R, |\mathbb N|<|A|<|\mathbb R|$ in your question  would have cardinality $\alpha-1$ if $\alpha$ is finite, and it would have cardinality $|\alpha|$ when $\alpha$ is infinite.
A: It depends. Do you care about naming or describing such sets?
In the system you propose, you may be able prove the existence of more than finitely many of them, but in a countable (non-infinitary, since you mention a small extension of ZFC) language, you will only be able to describe countable many of these sets or equivalence classes.
It may also be possible to show that there is no surjection from any given set with a cardinal (that you can describe or posit in this system) onto this class, but all that says is that the system you consider is too weak to analyze the class as fully as you want.  Back at you: if you were given an answer, what would you hope to do with it?
I am not expert enough to answer the question about consistency strength.  Based on the comments of Asaf and Monroe to the question, I suspect an assumption that the class you have is enumerated by a cardinal of type X is equivalent in strength to the assumption that a cardinal of type X exists.
Gerhard "Philosophers Don't Know What's Wanted" Paseman, 2019.04.27.
