Timeline for Is it consistent with ZFC that for all ordinals $\alpha, \beta < \omega$ it holds that $2^{\aleph_\alpha} = 2^{\aleph_\beta}$?

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Dec 6, 2011 at 15:44	comment	added	Emil Jeřábek		In any model of set theory, if $2^\kappa$ is constant for all regular $\kappa\le\gamma$, then it takes the same constant value at singulars below $\gamma$ since it is nondecreasing.
Dec 6, 2011 at 15:14	comment	added	Andreas Blass		It may also be worth noting that, if you use Easton's construction to get some desired values for $2^\kappa$ for all regular $\kappa$, then the values of $2^\kappa$ for singular $\kappa$ are automatically the smallest ones permitted by your choices for regular $\kappa$. In particular, if you make $2^\kappa$ constant for all regular $\kappa$ up to some chosen $\gamma$, then (in Easton models) $2^\kappa$ will also take the same constant value at singular cardinals below $\gamma$.
Dec 6, 2011 at 13:08	comment	added	Asaf Karagila♦		It is worth noting, for those unfamiliar with the theorem itself, that the resulting extension may also change the behaviour of $2^\kappa$ for $\kappa\notin F$ as well.
Dec 6, 2011 at 11:41	history	edited	Emil Jeřábek	CC BY-SA 3.0	need GCH in the ground model
Dec 6, 2011 at 11:05	history	answered	Emil Jeřábek	CC BY-SA 3.0