Pcf without choice Sh835

Saharon Shelah*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We mainly investigate models of set theory with restricted choice, e.g., ZF + DC + the family of countable subsets of λ is well ordered for every λ (really local version for a given λ). We think that in this frame much of pcf theory, (and combinatorial set theory in general) can be generalized. We prove here, in particular, that there is a proper class of regular cardinals, every large enough successor of singular is not measurable and we can prove cardinal inequalities. Solving some open problems, we prove that if μ>κ=cf(μ)>ℵ0, then from a well ordering of P(P(κ))∪κ>μ we can define a well ordering of κμ.

Original languageEnglish
Pages (from-to)623-654
Number of pages32
JournalArchive for Mathematical Logic
Volume63
Issue number5-6
DOIs
StatePublished - Jul 2024

Bibliographical note

Publisher Copyright:
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.

Keywords

  • 03E50
  • Pcf
  • Primary 03E17
  • Secondary: 03E05
  • Set theory
  • Weak axiom of choice

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