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- W4249397951 abstract "Abstract A truth for λ is a pair 〈 Q , ψ〉 so that Q ⊆ H λ , ψ is a first order formula with one free variable, and there exists B ⊆ H λ + such that ( H λ +; ∈, B ) ⊨ ψ[ Q ]. A cardinal λ is , indescribable just in case that for every truth 〈 Q , ψ〈 for λ, there exists < λ so that is a cardinal and 〈Q ∩ , ψ) is a truth for . More generally, an interval of cardinals [κ, λ] with κ ≤ λ is indescribable if for every truth 〈Q, ψ〈 for λ, there exists , and π: → H λ so that is a cardinal, is a truth for , and π is elementary from ( ) into ( H ; ∈, κ, Q ) with id. We prove that the restriction of the proper forcing axiom to ϲ-linked posets requires a indescribable cardinal in L, and that the restriction of the proper forcing axiom to ϲ + -linked posets, in a proper forcing extension of a fine structural model, requires a indescribable 1-gap [κ, κ + ]. These results show that the respective forward directions obtained in Hierarchies of Forcing Axioms I by Neeman and Schimmerling are optimal." @default.
- W4249397951 created "2022-05-12" @default.
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- W4249397951 date "2008-06-01" @default.
- W4249397951 modified "2023-09-26" @default.
- W4249397951 title "Hierarchies of forcing axioms II" @default.
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- W4249397951 doi "https://doi.org/10.2178/jsl/1208359058" @default.
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