A quasi-lower bound on the consistency strength of PFA

Abstract

A long-standing open question is whether supercompactness provides a lower bound on the consistency strength of the Proper Forcing Axiom (PFA). In this article we establish a quasi lower bound by showing that there is a model with a proper class of subcompact cardinals such that PFA (indeed the weaker statement that PFA holds for (2א0)+-linked forcings) fails in all of its proper forcing extensions. Neeman obtained such a result assuming the existence of “fine structural” models containing very large cardinals, however the existence of such models remains open. We show that Neeman’s arguments go through for a similar notion of “L-like” model and establish the existence of Llike models containing very large cardinals. The main technical result needed is the compatibility of Local Club Condensation with Acceptability in the presence of very large cardinals, a result which constitutes further progress in the outer model programme. The core model programme (initiated by Jensen, see Steel’s [16] for a survey) has had considerable success in establishing lower bounds on the consistency strength of set-theoretic statements, up to the level of Woodin cardinals. But the consistency strength of the Proper Forcing Axiom (PFA) is conjectured to be that of a supercompact cardinal, for which no core model theory is currently available. It is therefore worthwhile to consider quasi lower bounds on the consistency strength of PFA and the main result of this paper is that a proper class of subcompact cardinals serves as such a quasi lower bound: Theorem 1. Assuming the consistency of a proper class of subcompact cardinals, it is consistent that there is a proper class of subcompact cardinals, but PFA (even restricted to posets which are (2א0)+-linked) holds in no proper extension of the universe. What exactly is meant by a quasi lower bound? The necessary ingredients are • the desired set-theoretic principle φ for which we want to obtain a quasi-lower bound result 2000 Mathematics Subject Classification. 03E35, 03E55, 03E57.