Weak Indestructibility and Reflection

TL;DR

This paper explores the equiconsistency between weak indestructibility of certain cardinals and the existence of a proper class of strong reflecting strongs, using forcing and inner model techniques, and discusses connections to Woodin cardinals.

Contribution

It establishes an equiconsistency result linking weak indestructibility and reflection properties of strong cardinals, extending the understanding of their large cardinal strength.

Findings

Proves equiconsistency between weak indestructibility and strong reflection properties.

Extends weak indestructibility results beyond the next inaccessible limit.

Discusses connections between weak indestructibility and Woodin cardinals.

Abstract

This work is a part of my upcoming thesis [7]. We establish an equiconsistency between (1) weak indestructibility for all $\kappa +2$-degrees of strength for cardinals $\kappa $ in the presence of a proper class of strong cardinals, and (2) a proper class of cardinals that are strong reflecting strongs. We in fact get weak indestructibility for degrees of strength far beyond $\kappa +2$, well beyond the next inaccessible limit of measurables (of the ground model). One direction is proven using forcing and the other using core model techniques from inner model theory. Additionally, connections between weak indestructibility and the reflection properties associated with Woodin cardinals are discussed.