Possible Duplicate:
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...)

If ZFC+not(CH) is consistent, there should be sets of real numbers with cardinality strictly between N0 $\aleph_0$ and 2^N0. $2^{\aleph_0}$. 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...)

If ZFC+not(CH) is consistent, there should be sets of real numbers with cardinality strictly between N0 and 2^N0. 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...)