1) Wider research context.

This project focuses on set theory, particularly inner model theory and determinacy. The interplay between large cardinal axioms and determinacy axioms is a central theme in set theory, and recent advancements in the theory of hod mice have significantly driven progress in this area. The proposed research aims to fill crucial gaps in understanding (i) games of variable countable length, where the length of each run is countable but varies based on the moves, and (ii) models of the Axiom of Determinacy (AD) with structures not encodable as sets of reals.

2) Objectives.

The research is organized into three work packages.

(WP1) Long games and the analysis of Solovay-type models: Using generalized Solovay models introduced by Woodin, we will prove the level-by-level equivalences between the determinacy of games of variable countable length and the existence of inner models with large cardinals. We will also consider new Solovay-type models that satisfy much stronger forms of AD.

(WP2) Long games and unraveling methods: We will extend equivalence results from WP1 to the level of stronger large cardinals by making use of unraveling methods, which were first introduced by Marin's inductive proof of Borel determinacy.

(WP3) The analysis of Chang-type models: We will investigate new Chang-type models of determinacy derived from hod mice. An application of this work will yield novel results concerning generalized Chang models.

3) Methods.

All work packages will employ inner model theory extensively, and gaining deeper insights into the machinery is part of our motivation. The technical framework of the project builds on Neeman’s methods for long game determinacy, the analysis of HOD in models of AD, and the PI’s previous work on Chang-type models derived from hod mice and long game determinacy.

4) Level of innovation.

This project aims to discover new and intriguing classes of games of variable countable length. The determinacy of these games will form a much finer hierarchy of determinacy axioms than those currently known. We will also identify AD models associated with these games in WP1 and WP2. Combined with the method of core model induction, we will establish a theoretical framework for calibrating the exact consistency strength of various theories in the region of Woodin cardinals. Moreover, in WP3, we will produce even more complex AD models from hod mice. The Pmax extensions of these models are particularly compelling and lead to novel consistency results (e.g. recent work of Blue, Larson, and Sargsyan on the failures of square principles).

5) Primary researchers involved.

The PI is Takehiko Gappo and the mentor is Sandra Müller. The collaborators are Juan P. Aguilera, Nam Trang, and John R. Steel.