Laver Indestructibility and the Class of Compact Cardinals
Apter, Arthur W.
J. Symbolic Logic, Tome 63 (1998) no. 1, p. 149-157 / Harvested from Project Euclid
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.
Publié le : 1998-03-14
Classification: 
@article{1183745463,
     author = {Apter, Arthur W.},
     title = {Laver Indestructibility and the Class of Compact Cardinals},
     journal = {J. Symbolic Logic},
     volume = {63},
     number = {1},
     year = {1998},
     pages = { 149-157},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1183745463}
}
Apter, Arthur W. Laver Indestructibility and the Class of Compact Cardinals. J. Symbolic Logic, Tome 63 (1998) no. 1, pp.  149-157. http://gdmltest.u-ga.fr/item/1183745463/