Journal of Symbolic Logic

Laver Indestructibility and the Class of Compact Cardinals

Arthur W. Apter

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Abstract

Using an idea developed in joint work with Shelah, we show how to redefine Laver's notion of forcing making a supercompact cardinal $\kappa$ indestructible under $\kappa$-directed closed forcing to give a new proof of the Kimchi-Magidor Theorem in which every compact cardinal in the universe (supercompact or strongly compact) satisfies certain indestructibility properties. Specifically, we show that if K is the class of supercompact cardinals in the ground model, then it is possible to force and construct a generic extension in which the only strongly compact cardinals are the elements of K or their measurable limit points, every $\kappa \in K$ is a supercompact cardinal indestructible under $\kappa$-directed closed forcing, and every $\kappa$ a measurable limit point of K is a strongly compact cardinal indestructible under $\kappa$-directed closed forcing not changing $\wp(\kappa)$. We then derive as a corollary a model for the existence of a strongly compact cardinal $\kappa$ which is not $\kappa^+$ supercompact but which is indestructible under $\kappa$-directed closed forcing not changing $\wp(\kappa$) and remains non-$\kappa^+$ supercompact after such a forcing has been done.

Article information

Source
J. Symbolic Logic, Volume 63, Issue 1 (1998), 149-157.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183745463

Mathematical Reviews number (MathSciNet)
MR1610794

Zentralblatt MATH identifier
0899.03038

JSTOR
links.jstor.org

Citation

Apter, Arthur W. Laver Indestructibility and the Class of Compact Cardinals. J. Symbolic Logic 63 (1998), no. 1, 149--157. https://projecteuclid.org/euclid.jsl/1183745463


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