Hierarchies of forcing axioms I

Abstract. We prove new upper bound theorems on the consistency strengths of SPFA(), SPFA(-linked) and SPFA(+ -cc). Our results are in terms of (, )-subcompactness, which is a new large cardinal notion that combines the ideas behind subcompactness and -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 2 1-indescribable cardinal. Our upper bound for SPFA(c-c.c.) is a 2 2-indescribable cardinal, which is consistent with V = L. Our upper bound for SPFA(c+ -linked) is a cardinal that is (+ , 2 1)subcompact, which is strictly weaker than + -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 that is (+ , 2 2)-subcompact, which is also strictly weaker than + -supercompact.

• Type: Journal
• Year: 2008
• Where: JSYML
• Authors: Itay Neeman, Ernest Schimmerling
• Keywords: JSYML 2008, Large Cardinal Notion, Lower Bound, Upper Bound