Two remarks on Merimovich's model of the total failure of GCH

2021/02/01 by Mohammad Golshani, Golshani, Mohammad
Computer Science · Mathematics · #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.2102.00748

openalex publication_date 2021/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let M denote the Merimovich's model in which for each infinite cardinal λ, 2^λ=λ⁺³. We show that in M the following hold: (1) Shelah's strong hypothesis fails at all singular cardinals, indeed, ∀ λ(λ is a singular cardinal ⇒ pp(λ)=λ⁺³). (2) For each singular cardinal λ there is an inner model N of M such that M and N have the same bounded subsets of λ, λ is a singular cardinal in N, (λ⁺ⁱ)^N=(λ⁺ⁱ)^M, for i=1,2,3, and N \models 2^λ=λ⁺. Thus it is possible to add many new fresh subsets to λ without adding any new bounded subsets to λ.

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