Easton's Theorem in the presence of Woodin cardinals

Abstract

Under the assumption that $\delta$ is a Woodin cardinal and $\GCH$ holds, I show that if $F$ is any class function from the regular cardinals to the cardinals such that (1) $κ<\cf(F(κ))$, (2) $\kappa <\lambda$ implies $F(\kappa )\le F(\lambda )$, and (3) $\delta$ is closed under $F$, then there is a cofinality-preserving forcing extension in which $2^{\gamma }=F(\gamma )$ for each regular cardinal $\gamma <\delta$, and in which $\delta$ remains Woodin. Unlike the analogous results for supercompact cardinals [Men76] and strong cardinals [FH08], there is no requirement that the function $F$ be locally definable.

22 pages

References in corpus (1)

• Some results on large cardinals and the continuum function