Hierarchies of forcing axioms I

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(cc.c.) is a cardinal κ that is (κ+,Σ22)-subcompact, which is also strictly weaker than κ-supercompact.