TY - JOUR

T1 - On a class of maximality principles

AU - Ikegami, Daisuke

AU - Trang, Nam

N1 - Funding Information:
Acknowledgements The paper was written during the second author’s visit to Kobe University, where the first author was a postdoctoral researcher in May 2014 and completed during the first author’s visit to UC Irvine, where the second author is a Visiting Assistant Professor, in August 2016. The first author would like to thank Toshimichi Usuba for many comments and discussions on this topic. He is also grateful for JSPS for support through the grants with JSPS KAKENHI Grant Numbers 14J02269 and 15K17586. The second author would like to thank the NSF for its generous support through Grant DMS-1565808. Finally, we would like to thank Gunter Fuchs, Kaethe Minden, and the anonymous referee for several helpful comments regarding the content of the paper.

PY - 2018/8/1

Y1 - 2018/8/1

N2 - 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.

AB - 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.

KW - Forcing axioms

KW - Inner models

KW - Large cardinals

KW - Maximality principles

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U2 - 10.1007/s00153-017-0603-2

DO - 10.1007/s00153-017-0603-2

M3 - Article

AN - SCOPUS:85035128059

VL - 57

SP - 713

EP - 725

JO - Archive for Mathematical Logic

JF - Archive for Mathematical Logic

SN - 0933-5846

IS - 5-6

ER -