Nairian Models
Douglas Blue, Paul B. Larson, and Grigor Sargsyan

TL;DR
This paper introduces Nairian models, a hierarchy of models of the Axiom of Determinacy, and uses them to derive results about forcing, large cardinals, and obstructions in inner model theory.
Contribution
It constructs new Nairian models to explore the limits of inner model theory and forcing, providing answers to open questions about large cardinals and iterability.
Findings
Constructed models satisfying ZFC+MM++(c)+ negations of square principles.
Demonstrated the failure of the Iterability Conjecture in certain models.
Provided a negative answer to a question about the equiconsistency of supercompactness and Woodin cardinals.
Abstract
We introduce a hierarchy of models of the Axiom of Determinacy called \emph{Nairian models}. Forcing over the simplest Nairian model, we obtain a model of . Then, fixing , we design a Nairian model and force over it to produce a model of . We also build a Nairian model that satisfies is a supercompact cardinal." We obtain as corollaries of these constructions (1) the consistent failure of the Iterability Conjecture for the Mitchell-Schindler construction, (2) the consistent failure of the Iterability Conjecture for the construction using -complete (for any finite stack of exponents) background extenders, answering a strong version of a question…
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Taxonomy
TopicsAdvanced Topology and Set Theory · Homotopy and Cohomology in Algebraic Topology · Computability, Logic, AI Algorithms
