On a class of maximality principles

Daisuke Ikegami, Nam Trang

Research output: Contribution to journalArticle

Abstract

We study various classes of maximality principles, MP (κ, Γ) , introduced by Hamkins (J Symb Log 68(2):527–550, 2003), where Γ defines a class of forcing posets and κ is an infinite cardinal. We explore the consistency strength and the relationship of MP(κ, Γ) with various forcing axioms when κ∈ { ω, ω1}. In particular, we give a characterization of bounded forcing axioms for a class of forcings Γ in terms of maximality principles MP(ω1, Γ) for Σ 1 formulas. A significant part of the paper is devoted to studying the principle MP(κ, Γ) where κ∈ { ω, ω1} and Γ defines the class of stationary set preserving forcings. We show that MP(κ, Γ) has high consistency strength; on the other hand, if Γ defines the class of proper forcings or semi-proper forcings, then by Hamkins (2003), MP(κ, Γ) is consistent relative to V= L.

Original languageEnglish
Pages (from-to)713-725
Number of pages13
JournalArchive for Mathematical Logic
Volume57
Issue number5-6
DOIs
Publication statusPublished - 2018 Aug 1
Externally publishedYes

Keywords

  • Forcing axioms
  • Inner models
  • Large cardinals
  • Maximality principles

ASJC Scopus subject areas

  • Philosophy
  • Logic

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