Hierarchies of forcing axioms II

A Σ21 truth for λ is a pair 〈Q,ψ〉 so that Q ⊆ Hλ, 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 Σ21 indescribable just in case that for every Σ 2 1 truth 〈Q,ψ〉 for λ, there exists λ̄ < λ so that λ̄ is a cardinal and 〈Q ∩ Hλ̄, ψ〉 is a Σ 2 1 truth for λ̄. More generally, an interval of cardinals [κ, λ] with κ ≤ λ is Σ21 indescribable if for every Σ 2 1 truth 〈Q,ψ〉 for λ, there exists κ̄ ≤ λ̄ < κ, Q̄ ⊆ Hλ̄, and π : Hλ̄ → Hλ so that λ̄ is a cardinal, 〈Q̄, ψ〉 is a Σ21 truth for λ̄, and π is elementary from (Hλ̄;∈, κ̄, Q̄) into (Hλ;∈, κ,Q) with π↾ κ̄ = id. We prove that the restriction of the proper forcing axiom to c-linked posets requires a Σ21 indescribable cardinal in L, and that the restriction of the proper forcing axiom to c-linked posets, in a proper forcing extension of a fine structural model, requires a Σ21 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. It is a well-known conjecture that the large cardinal consistency strength of PFA is a supercompact cardinal. This paper is the second in a pair of papers connecting a hierarchy of forcing axioms leading to PFA with a hierarchy of large cardinal axioms leading to supercompact. Recall that a forcing notion P is λ-linked if it can be written as a union of sets Pξ, ξ < λ, so that for each ξ, the conditions in Pξ are pairwise compatible. PFA(λ-linked) is the restriction of PFA to λ-linked posets. The forcing axioms form a hierarchy, with PFA of course equivalent to the statement that PFA(λlinked) holds for all λ. The following theorem deals with consistency strength at the low end of this hierarchy. Theorem A. The consistency strength of PFA(c-linked) is precisely a Σ1 indescribable cardinal. More specifically: 1. If κ is Σ1 indescribable in a model M satisfying the GCH then there is a forcing extension of M , by a proper poset, in which c = ω2 = κ and PFA(c-linked) holds. 2. If PFA(c-linked) holds then (ω2) V is Σ1 indescribable in L. The statement that c = ω2 in part (1) is redundant, as PFA(c-linked) implies c = ω2. This was proved by Todorčević (see Bekkali [2]) and Veličković [15]. Part (1) is joint with Schimmerling: Neeman–Schimmerling [7] proves its semi-proper analogue, producing a semiproper forcing extension of M in which SPFA(c-linked) holds, and the proof of (1) is identical except for the routine change of replacing semi-proper by proper throughout. It follows from part (2) This material is based upon work supported by the National Science Foundation under Grant No. DMS-0556223