### 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, ω_{1}}. In particular, we give a characterization of bounded forcing axioms for a class of forcings Γ in terms of maximality principles MP(ω_{1}, Γ) for Σ _{1} formulas. A significant part of the paper is devoted to studying the principle MP(κ, Γ) where κ∈ { ω, ω_{1}} 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 language | English |
---|---|

Pages (from-to) | 713-725 |

Number of pages | 13 |

Journal | Archive for Mathematical Logic |

Volume | 57 |

Issue number | 5-6 |

DOIs | |

Publication status | Published - 2018 Aug 1 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Philosophy
- Logic

### Cite this

*Archive for Mathematical Logic*,

*57*(5-6), 713-725. https://doi.org/10.1007/s00153-017-0603-2

**On a class of maximality principles.** / Ikegami, Daisuke; Trang, Nam.

Research output: Contribution to journal › Article

*Archive for Mathematical Logic*, vol. 57, no. 5-6, pp. 713-725. https://doi.org/10.1007/s00153-017-0603-2

}

TY - JOUR

T1 - On a class of maximality principles

AU - Ikegami, Daisuke

AU - Trang, Nam

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

UR - http://www.scopus.com/inward/record.url?scp=85035128059&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85035128059&partnerID=8YFLogxK

U2 - 10.1007/s00153-017-0603-2

DO - 10.1007/s00153-017-0603-2

M3 - Article

VL - 57

SP - 713

EP - 725

JO - Archive for Mathematical Logic

JF - Archive for Mathematical Logic

SN - 0933-5846

IS - 5-6

ER -