J-2 - Consistency Results in Topology, I: Quotable Principles

Abstract

Consistency results in topology are obtained in two ways. The first way consists of proving implications among statements, where the antecedent is a statement previously proven consistent with Zermelo–Fraenkel set theory with the axiom of choice (ZFC). Thus, the consequent itself is proven consistent with ZFC as well. This chapter highlights results like these and, in particular, the best-known combinatorial principles that occur as antecedents. The second kind of consistency results requires an intimate knowledge of the way in which (relative) consistency results are actually proven. The best-known quotable principle is undoubtedly the Continuum Hypothesis (CH), which states that the real line has minimum possible cardinality. Every consequence of CH is a potential theorem, as it can no longer be refuted. Some consequences are equivalent to CH—for example, the existence of the special infinite-dimensional space; such consequences are then automatically non provable because Cohen proved the consistency of the negation of CH. To decide other consequences, proofs that avoid CH or principles that imply their negations are needed. Martin's Axiom is such a principle. It can be viewed as an extension of the Baire Category Theorem.