Closure properties of measurable ultrapowers

TL;DR

This paper investigates the closure properties of measurable ultrapowers in set theory, showing their dependence on the underlying model's combinatorial features and establishing results about their consistency strength.

Contribution

It demonstrates how the closure properties of measurable ultrapowers vary with set-theoretic assumptions and develops techniques to analyze their consistency strength.

Findings

Strongly compact cardinals can produce ultrapowers with maximal closure properties.

Canonical inner models' ultrapowers have minimal closure properties due to square principles.

Techniques link closure properties to the consistency strength of set-theoretic hypotheses.

Abstract

We study closure properties of measurable ultrapowers with respect to Hamkin's notion of "freshness" and show that the extent of these properties highly depends on the combinatorial properties of the underlying model of set theory. In one direction, a result of Sakai shows that, by collapsing a strongly compact cardinal to become the double successor of a measurable cardinal, it is possible to obtain a model of set theory in which such ultrapowers possess the strongest possible closure properties. In the other direction, we use various square principles to show that measurable ultrapowers of canonical inner models only possess the minimal amount of closure properties. In addition, the techniques developed in the proofs of these results also allow us to derive statements about the consistency strength of the existence of measurable ultrapowers with non-minimal closure properties.