Easton functions and supercompactness

Brent Cody, Sy-David Friedman, Radek Honzik

Published 2013-11-01Version 1

Suppose $\kappa$ is $\lambda$-supercompact witnessed by an elementary embedding $j:V\rightarrow M$ with critical point $\kappa$, and further suppose that $F$ is a function from the class of regular cardinals to the class of cardinals satisfying the requirements of Easton's theorem: (1) $\forall\alpha$ $\alpha<\textrm{cf}(F(\alpha))$ and (2) $\alpha<\beta$ $\Longrightarrow$ $F(\alpha)\leq F(\beta)$. In this article we address the question: assuming GCH, what additional assumptions are necessary on $j$ and $F$ if one wants to be able to force the continuum function to agree with $F$ globally, while preserving the $\lambda$-supercompactness of $\kappa$? We show that, assuming GCH, if $F$ is any function as above, and in addition for some regular cardinal $\lambda>\kappa$ there is an elementary embedding $j:V\rightarrow M$ with critical point $\kappa$ such that $\kappa$ is closed under $F$, the model $M$ is closed under $\lambda$-sequences, $H(F(\lambda))\subseteq M$, and for each regular cardinal $\gamma\leq \lambda$ one has $(|j(F)(\gamma)|=F(\gamma))^V$, then there is a cardinal-preserving forcing extension in which $2^\delta=F(\delta)$ for every regular cardinal $\delta$ and $\kappa$ remains $\lambda$-supercompact. This answers a question of B. Cody, M. Magidor, On supercompactness and the continuum function, Ann. Pure Appl. Logic, (2013).