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Cardinal arithmetic does not seem to open its way to the existence of $\aleph_1$ that is not $2^{\aleph_0}$, as any operation on $\aleph_0$ would lead to $\aleph_0$ or $2^{\aleph_0}$ and $2^{2^{\aleph_0}}$ or so forth.

According to my knowledge, choice does not determine whether continuum hypothesis is right or wrong, so what is going on with this?

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    To add to the answers below, there are many operations in cardinal arithmetic that are more subtle than talking products or powers. (To mention just one example, you may want to look at the smallest size of a covering (or cofinal) family of countable subsets of $A$ rather than to $A^{\aleph_0}$.) These operations typically give you values undecided by the usual axioms, and one applied to $\aleph_0$, may very well give you the value $\omega_1$ in some models, regardless of what $2^{\aleph_0}$ is in these models.2012-09-23
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    (than *taking* products)2012-09-28

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