2025/01/31 by Douglas Blue, Paul Larson, Blue, Douglas +3
Mathematics · Computer Science · #Advanced Topology and Set Theory #Homotopy and Cohomology in Algebraic Topology #Computability, Logic, AI Algorithms
paper · pdf · doi:10.48550/arxiv.2501.18958
We introduce a hierarchy of models of the Axiom of Determinacy called Nairian models. Forcing over the simplest Nairian model, we obtain a model of \sfZFC+\sfMM++(c)+¬\squareω3+¬\square(ω3). Then, fixing n∈ [3, ω), we design a Nairian model and force over it to produce a model of \sfZFC+\sfMM++(c)+∀ i∈ [2, n] ¬\square(ωi). We also build a Nairian model that satisfies \sfZF+"ω1 is a supercompact cardinal." We obtain as corollaries of these constructions (1) the consistent failure of the Iterability Conjecture for the Mitchell-Schindler \sfKc construction, (2) the consistent failure of the Iterability Conjecture for the \sfKc construction using 2^2… 2ω-complete (for any finite stack of exponents) background extenders, answering a strong version of a question asked by Steel, and (3) a negative answer to Trang's question whether \sfZF+"ω1 is a supercompact cardinal" is equiconsistent with \sfZFC+"there is a proper class of Woodin cardinals that are limits of Woodin cardinals." These corollaries identify obstructions to extending the methods of (descriptive) inner model theory past a Woodin cardinal which is a limit of Woodin cardinals.