Indestructibility, strong compactness, and level by level equivalence

Volume 204 / 2009

Arthur W. Apter Fundamenta Mathematicae 204 (2009), 113-126 MSC: 03E35, 03E55. DOI: 10.4064/fm204-2-2


We show the relative consistency of the existence of two strongly compact cardinals $\kappa _1$ and $\kappa _2$ which exhibit indestructibility properties for their strong compactness, together with level by level equivalence between strong compactness and supercompactness holding at all measurable cardinals except for $\kappa _1$. In the model constructed, $\kappa _1$'s strong compactness is indestructible under arbitrary $\kappa _1$-directed closed forcing, $\kappa _1$ is a limit of measurable cardinals, $\kappa _2$'s strong compactness is indestructible under $\kappa _2$-directed closed forcing which is also $(\kappa _2, \infty )$-distributive, and $\kappa _2$ is fully supercompact.


  • Arthur W. ApterDepartment of Mathematics

    Baruch College of CUNY

    New York, NY 10010, U.S.A.
    The CUNY Graduate Center, Mathematics

    365 Fifth Avenue

    New York, NY 10016, U.S.A.

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