Blowing up the power of a singular cardinal of uncountable cofinality with collapses

2020/11/01 by Sittinon Jirattikansakul, Jirattikansakul, Sittinon
Computer Science · Economics, Econometrics and Finance · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Economic theories and models #FOS: Mathematics #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.2011.00409

openalex publication_date 2020/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The \em Singular Cardinal Hypothesis (SCH) is one of the most classical combinatorial principles in set theory. It says that if κ is singular strong limit, then 2^κ=κ⁺. We prove that given a singular cardinal κ of \em cofinality η in the ground model, which is a limit of suitable large cardinals, and η⁺=ℵ_γ, then there is a forcing extension which preserves cardinals and cofinalities up to and including η, such that κ becomes ℵ_γ+η, and SCH fails at κ. Furthermore, if η is not an ℵ-fixed point, then in our model, SCH fails at ℵ_η. Our large cardinal assumption is below the existence of a Woodin cardinal. In our model we also obtain a very good scale.

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