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Fundamenta Mathematicae

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Easton functions and supercompactness

Tom 226 / 2014

Fundamenta Mathematicae 226 (2014), 279-296 MSC: Primary 03E35; Secondary 03E55. DOI: 10.4064/fm226-3-6

Streszczenie

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].

Autorzy

• Brent CodyDepartment of Mathematics
and Applied Mathematics
Virginia Commonwealth University
1015 Floyd Avenue
Richmond, VA 23284, U.S.A.
e-mail
• Sy-David FriedmanKurt Gödel Research Center
for Mathematical Logic
University of Vienna
Währinger Straße 25
1090 Wien, Austria
e-mail
• Radek HonzikKurt 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
e-mail

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