The Proper Forcing Axiom, Prikry forcing, and the Singular Cardinals Hypothesis

Abstract

The purpose of this paper is to present some results which suggest that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. What will be proved is that a form of simultaneous reflection follows from the Set Mapping Reflection Principle, a consequence of PFA. While the results fall short of showing that MRP implies SCH, it will be shown that MRP implies that if SCH fails first at κ then every stationary subset of S κ + ω = { α < κ + : cf ( α ) = ω } reflects. It will also be demonstrated that MRP always fails in a generic extension by Prikry forcing.