## Journal of Symbolic Logic

### Laver Indestructibility and the Class of Compact Cardinals

Arthur W. Apter

#### 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

https://projecteuclid.org/euclid.jsl/1183745463

Mathematical Reviews number (MathSciNet)
MR1610794

Zentralblatt MATH identifier
0899.03038

JSTOR