Easton functions and supercompactness

Brent Cody¹ ; Sy-David Friedman² ; Radek Honzik³

¹ Department of Mathematics and Applied Mathematics Virginia Commonwealth University 1015 Floyd Avenue Richmond, VA 23284, U.S.A.
² Kurt Gödel Research Center for Mathematical Logic University of Vienna Währinger Straße 25 1090 Wien, Austria
³ Kurt Gödel Research Center for Mathematical Logic University of Vienna Währinger Straße 25 1090 Wien, Austria and Department of Logic Charles University Palachovo nám. 2 116 38 Praha 1, Czech Republic

Fundamenta Mathematicae, Tome 226 (2014) no. 3, pp. 279-296

Résumé

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].

Détail

DOI : 10.4064/fm226-3-6

Keywords: suppose kappa lambda supercompact witnessed elementary embedding critical point kappa further suppose function class regular cardinals class cardinals satisfying requirements eastons theorem forall alpha alpha mathop alpha alpha beta rightarrow alpha leq beta address question assuming gch what additional assumptions necessary wants able force continuum function agree globally while preserving lambda supercompactness kappa assuming gch function above addition regular cardinal lambda kappa there elementary embedding critical point kappa kappa closed under model closed under lambda sequences lambda subseteq each regular cardinal gamma leq lambda has gamma gamma there cardinal preserving forcing extension which delta delta every regular cardinal delta kappa remains lambda supercompact answers question