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

A noteworthy feature of general topology – as contrasted with geometry, number theory, and most other fields of mathematics – is that many of its fundamental questions are not decided by " the usual axioms of set theory ". What this means is that if one codifies the set-theoretic assumptions used implicitly in ordinary mathematics and consolidates these in a list of axioms – the usual axioms of set theory – one is left with many statements that can neither be proved nor refuted on the basis of these axioms alone. In some sense such undecidability results are commonplace: everyone knows that, say, the statement ϕ = (∀x)(∀y)(x y = yx) cannot be decided by " the usual axioms of Group Theory ". Indeed, the group of integers is a model for " Group Theory plus ϕ " , which makes this conjunction consistent and ϕ non-refutable; the permutation group S 3 does the same for ¬ϕ. A convenient list of axioms for Set Theory, one which has become standard is " ZFC " – Zermelo–Fraenkel set theory , including the Axiom of Choice. See, e.g., [Ku] for a detailed exposition. Thus there are important topological statements ϕ such that neither ϕ nor its negation follow from ZFC. Establishing this is not as easy as in the case of Group Theory. While it is easy to prove the consistency of Group Theory by providing models (groups) for it, the same is, by Gödel's Second Incompleteness Theorem, in principle impossible for ZFC (or any other useful collection of axioms for set theory). Gödel's theorem says that a consistency proof of a theory as strong as ZFC cannot be formalized in the theory itself. This explains why many results are formulated as " if ZFC is consistent then so is ZFC plus ϕ " instead of simply " ϕ is consistent with ZFC " and why, formally, we should be speaking of relative consistency proofs. Even if one is not interested in consistency results per se, it is nonetheless prudent to be aware of them, lest one waste effort trying to prove a proposition that has a consistent negation. There are two ways in which consistency results in topo-logy are obtained. The first consists of proving implications between statements, where the antecedent is a statement previously proven consistent with ZFC. Thus, the consequent itself is proven consistent with ZFC as well. The …