Indestructibility, strong compactness, and level by level equivalence

Arthur W. Apter ¹

¹ Department of Mathematics Baruch College of CUNY New York, NY 10010, U.S.A. and The CUNY Graduate Center, Mathematics 365 Fifth Avenue New York, NY 10016, U.S.A.

Fundamenta Mathematicae, Tome 204 (2009) no. 2, pp. 113-126

Résumé

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.

Détail

DOI : 10.4064/fm204-2-2

Keywords: relative consistency existence strongly compact cardinals kappa kappa which exhibit indestructibility properties their strong compactness together level level equivalence between strong compactness supercompactness holding measurable cardinals except kappa model constructed kappa strong compactness indestructible under arbitrary kappa directed closed forcing kappa limit measurable cardinals kappa strong compactness indestructible under kappa directed closed forcing which kappa infty distributive kappa fully supercompact