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There is a famous quote of Paul Cohen which reads $\lt\lt$ A point of view which the author feels may eventually come to be accepted is that the continuum hypothesis is obviously false ... The continuum $\mathfrak c$ is greater than $\aleph_n,\aleph_\omega,\aleph_{\alpha}$ where $\alpha=\aleph_\omega$ etc. This point of view regards $\mathfrak c$ as an incredibly rich set given to us by one bold new axiom (...) $\gt\gt$

Paul Cohen's opinion implies that there is a weakly inacessible cardinal below $\mathfrak c$. Let us denote WIBC (weakly inaccessible below continuum) this hypothesis (does that hypothesis already have a name in the literature?). Since the existence of a weakly inacessible cardinal cannot be shown to be consistent with $ZFC$, we will never be able prove that WIBC is consistent with ZFC (unless of course ZFC is inconsistent).

But if we assume that a weakly inacessible cardinal exists, can one use a variation of Easton's forcing method to show that WIBC is consistent with ZFC?

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    @Asaf : Glad to hear it's not a stupid question!2011-10-20

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You don't need all of the Easton machinery to do this, as simple Cohen forcing will do the job. For example, one might start with an inaccessible cardinal $\kappa$ in $L$, and forcing with the finite partial functions from $\kappa$ to $\{0,1\}$ (i.e., "adding $\kappa$ Cohen reals") will result in a model where $2^{\aleph_0}=\kappa$. This forcing is mild, in that it preserves cofinalities, so $\kappa$ will still be a regular limit cardinal in the extension.

Similar arguments (assuming the existence of the appropriate cardinals in $L$) that the continuum can be weakly Mahlo, etc., so one can have lots of weakly inaccessible cardinals below $2^{\aleph_0}$ given only mild large cardinal assumptions.

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    Welcome to math.SE!2011-10-27