Easton functions and supercompactness
Volume 226 / 2014
Abstract
Suppose that $\kappa $ is $\lambda $-supercompact witnessed by an elementary embedding $j:V\to M$ 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 $ $\alpha <\mathop {\rm cf}(F(\alpha ))$, and (2) $\alpha <\beta \Rightarrow F(\alpha )\leq F(\beta )$. We address the question: assuming ${\rm GCH}$, 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 ${\rm GCH}$, if $F$ is any function as above, and in addition for some regular cardinal $\lambda >\kappa $ there is an elementary embedding $j:V\to M$ 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 [CM14].
