Partial Tower Sealing

TL;DR

This paper investigates the conditions under which a weak form of Tower Sealing holds in certain set-theoretic models, revealing its limitations and implications for large cardinal hypotheses.

Contribution

It introduces and analyzes Partial Tower Sealing, a weaker form of Tower Sealing, and establishes its consistency strength relative to large cardinal assumptions.

Findings

Partial Tower Sealing holds in some generic extensions of hod mice with strong cardinals.

Full Tower Sealing fails in these extensions.

Partial Tower Sealing implies Sealing and has a lower consistency strength than ZFC with a Woodin limit of Woodin cardinals.

Abstract

The main result of this paper shows that a weak form of Tower Sealing holds in a generic extension of hod mice with a strong cardinal and a proper class of Woodin cardinals. We show Tower Sealing fails in such extensions in general. We show that this weak form of Tower Sealing (called Partial Tower Sealing) implies Sealing and that its consistency strength is below that of ZFC + there is a Woodin limit of Woodin cardinals.