Indestructibility of Weakly Compact Cardinals
by
Thomas Johnstone
City University of New York

Large Cardinals may get destroyed or stay preserved after forcing with a particular poset. It has been of major interest in modern set theory to investigate which cardinals can be made indestructible by various classes of forcing. Such theorems often provide a way of obtaining new independence results.

While lots of research has dealt with cardinals stronger than measurable cardinals, my focus in this talk will be on cardinals weaker than measurable cardinals, in particular weakly compact cardinals and strongly unfoldable cardinals. Both are consistent with V=L and resemble miniature measurable cardinals. Strongly unfoldable cardinals have in addition some of the flavor of strong cardinals.

Given a strongly unfoldable cardinal, I will provide a forcing extension in which the cardinal is indestructible by all kappa-closed, kappa-proper forcing. In comparison to Richard Laver's celebrated indestructibility result for supercompact cardinals (1978), this result reduces the large cardinal hypothesis significantly but at the expense of less indestructibility.

Date received: January 16, 2006