J-3 - Consistency Results in Topology, II: Forcing and Large Cardinals

Abstract

This chapter describes the forcing method, which is well known in topology. This method works fairly well when the problems involve sets of bounded cardinality but it tends to fail in certain conditions. A prime example is the Normal Moore Space Conjecture, which cannot be proved consistent assuming the consistency of Zermelo–Fraenkel set theory with the axiom of choice (ZFC) alone—the proof requires additional, stronger axioms that most set theorists regard as safe to assume. Such axioms assert the (consistency of the) existence of large cardinals. A cardinal number is large if the assumption of its existence, when added to the axioms of ZFC, proves the consistency of ZFC. Finer analyses of large cardinals consider a hierarchy of consistency strength, that is, the consistency of a cardinal with property P implies the consistency of a cardinal with property. The chapter also discusses some of the most useful models employed in consistency results, equivalently, the most useful partial orders.