Easton's Theorem in the presence of Woodin cardinals

TL;DR

This paper extends Easton's theorem to models with Woodin cardinals, showing that continuum function values can be freely assigned below a Woodin cardinal while preserving its properties.

Contribution

It demonstrates that Easton's theorem holds in the presence of Woodin cardinals without requiring local definability of the continuum function.

Findings

Existence of a cofinality-preserving extension with specified continuum function values

Preservation of Woodin cardinal properties in the forcing extension

No need for local definability of the continuum function in this context

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) $\kappa<\cf(F(\kappa))$, (2) $\kappa<\lambda$ implies $F(\kappa)\leq F(\lambda)$, 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.