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Easton functions and supercompactness

Tom 226 / 2014

Brent Cody, Sy-David Friedman, Radek Honzik 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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