Some combinatorial properties of Ultimate L and V

TL;DR

This paper explores the structure of large cardinals under strong compactness assumptions, providing evidence for the Ultimate L Conjecture and answering several open questions in set theory.

Contribution

It establishes new constraints on large cardinals, relates them to the Ultrapower Axiom, and advances understanding of inner models and ultrafilters in set theory.

Findings

Regular cardinals above the first strongly compact with indecomposable ultrafilters are measurable.

Successor almost strongly compact cardinals of uncountable cofinality are strongly compact.

If a proper class of strongly compact cardinals exists, there are no nontrivial elementary embeddings from V to inner models.

Abstract

This paper establishes a number of constraints on the structure of large cardinals under strong compactness assumptions. These constraints coincide with those imposed by the Ultrapower Axiom, a principle that is expected to hold in Woodin's hypothesized Ultimate \(L\), providing some evidence for the Ultimate \(L\) Conjecture. We show that every regular cardinal above the first strongly compact that carries an indecomposable ultrafilter is measurable, answering a question of Silver for large enough cardinals. We show that any successor almost strongly compact cardinal of uncountable cofinality is strongly compact, making progress on a question of Boney, Unger, and Brooke-Taylor. We show that if there is a proper class of strongly compact cardinals then there is no nontrivial cardinal preserving elementary embedding from the universe of sets into an inner model, answering a question of Caicedo granting large cardinals. Finally, we show that if \(\kappa\) is strongly compact, then \(V\) is a set forcing extension of the inner model \(\kappa\text{-HOD}\) consisting of sets that are hereditarily ordinal definable from a \(\kappa\)-complete ultrafilter over an ordinal; \(\kappa\text{-HOD}\) seems to be the first nontrivial example of a ground of \(V\) whose definition does not involve forcing.