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?




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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? 



Cardinals in $ZFC+\neg CH$
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.


