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In set theories where Continuum Hypothesis is false, what are the new sets?

If ZFC+not(CH) is consistent, there should be sets of real numbers with cardinality strictly between $\aleph_0$ and $2^{\aleph_0}$. Then why hasn't someone constructed a set of real numbers of intermediate cardinality in a model of ZFC+not(CH)?

(I assume there are good reasons why this would be hard, so I'm asking what those reasons are rather than suggesting that I've come up with a new angle of attack...)

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 Because they have. set $= \mathbb{R}\cap L \in \operatorname{force}(L,(\omega_2)_L$ random reals$)$ – Ricky Demer Dec 6 2010 at 3:48
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 Asked and answered: mathoverflow.net/questions/10227/… – Nate Eldredge Dec 6 2010 at 3:49
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 "why hasn't someone constructed a set of reals of intermediate cardinality" This is simply false. Many constructions give models with sets of reals of intermediate size. How explicit they are may depend on the combinatorics of the model in question, on how the model was built, on the large cardinals or inner models of the model, etc. I assume perhaps your question is why there is no canonical set of intermediate cardinality? – Andres Caicedo Dec 6 2010 at 3:51
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 See also this question mathoverflow.net/questions/28806/… and its answers for a close parallel to the current question. – Joel David Hamkins Dec 6 2010 at 14:39
I saw this and then looked at that other question, and posted a new answer. – Michael Hardy Dec 6 2010 at 18:22

closed as exact duplicate by Nate Eldredge, Will Jagy, Gerry Myerson, Pete L. Clark, S. Carnahan♦ Dec 6 2010 at 4:46

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