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Is it possible to create a model of ZFC, so that the cardinality of each power set is a limit cardinal (as opposed to GCH where they are always successor cardinals)?

Take for example the following rule (which is the one that gives the smallest limit cardinal possible for regular cardinals because of the cofinality requirement): $$2^{\aleph_\alpha} = \aleph_{\aleph_{\alpha + 1}}$$ Due to Easton's theorem, we can build a model where this rule applies to all regular cardinal numbers.

If we want to apply it to singular cardinals, we must be mindful of several more rules (see 3 in http://www.math.tau.ac.il/~gitik/icm5.pdf). But as far as I can tell, all the known rules are satisfied, including Shelah's upper limit for the power set of strong limit cardinals: in this model they must be also fixed points of the $\aleph$ function, so we get $2^{\aleph_\delta} = \aleph_{\aleph_{\delta + 1}} = \aleph_{\aleph_\delta^+} = \aleph_{\delta^+} < \aleph_{{|\delta|}^{+4}}$.

So is this kind of thing possible?

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  • $\begingroup$ Without assuming GCH, $\aleph_1$ is not equal to c. $\endgroup$
    – Alon Navon
    Dec 23, 2015 at 20:48
  • $\begingroup$ @RossMillikan It is consistent that $\mathfrak{c}$ is not regular - it just has to have uncountable cofinality. $\endgroup$
    – Noah Schweber
    Dec 23, 2015 at 20:49

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Turns out this is possible assuming a strong cardinal. For the complete solution by Mohammad Golshani, see https://mathoverflow.net/questions/226887/when-can-power-sets-be-limit-cardinals.

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