Forcing "NS_ω1 is ω₁-dense" From Large Cardinals

2024/03/14 by Andreas Lietz, Lietz, Andreas
Physics and Astronomy · #03E57 03E55 03E50 03E35 #Cosmology and Gravitation Theories #FOS: Mathematics #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.2403.09020

openalex publication_date 2024/03/14 · openalex created_date 2024/03/16 · openalex updated_date 2026/07/28

Abstract

We answer a question of Woodin by showing that assuming an inaccessible cardinal κ which is a limit of &lt;κ-supercompact cardinals exists, there is a stationary set preserving forcing ℙ so that V^{\mathbb P}\models``NS<sub>ω1</sub> is ω<sub>1</sub>-dense". We also introduce a new forcing axiom QM, show it is consistent assuming a supercompact limit of supercompact cardinals and prove that it implies ℚ<sub>max</sub>-(*). Consequently, QM implies ``NS_ω1 is ω₁-dense".

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