Hierarchies of forcing axioms I

Abstract

We prove new upper bound theorems on the consistency strengths of SPFA(θ), SPFA(θ-linked) and SPFA(θ⁺-cc). Our results are in terms of (θ,Γ)-subcompactness, which is a new large cardinal notion that combines the ideas behind subcompactness and Γ-indescribability. Our upper bound for SPFA(𝔠-linked) has a corresponding lower bound, which is due to Neeman and appears in his follow-up to this paper. As a corollary, SPFA(𝔠-linked) and PFA(𝔠-linked) are each equiconsistent with the existence of a Σ²₁-indescribable cardinal. Our upper bound for SPFA(𝔠-c.c.) is a Σ²₂-indescribable cardinal, which is consistent with V=L. Our upper bound for SPFA(𝔠⁺-linked) is a cardinal κ that is (κ⁺, Σ²₁)-subcompact, which is strictly weaker than κ⁺-supercompact. The axiom MM(𝔠) is a consequence of SPFA(𝔠⁺-linked) by a slight refinement of a theorem of Shelah. Our upper bound for SPFA(𝔠⁺⁺-c.c.) is a cardinal κ that is (κ⁺, Σ²₂)-subcompact, which is also strictly weaker than κ⁺-supercompact.