Abstract

Is the Axiom of Foundation a truth or a falsehood about the universe of sets? Such questions have always struck this reviewer as unmathematical and a bit silly. Of course, that appraisal is more like a grunt of disapproval than an argument. We would welcome arguments. Justin Clarke-Doane offers a pretty good one that goes roughly as follows.Mathematicians making reasoned judgments about set-theoretic principles sometimes take positions that appear incompatible: one mathematician making a claim that would contradict the claim of another if both claims were treated as assertions about a single universe of sets. If the claims are sincerely made and contradict each another, one party to the dispute must be expressing a false belief. It is characteristic of reliable methods that they significantly impede the formation of false beliefs. Reliable methods for fixing set-theoretic beliefs significantly impede the formation of false beliefs about sets. So, if we knew which party in a set-theoretic dispute has got it wrong, we would have evidence that that party’s methods are unreliable. But it is frequently unclear why the reasons offered by one side should prevail over those offered by the other. Unable to pinpoint the error, we could find ourselves less confident about the methods of each party. We might even wonder if there is much hope of establishing the reliability of set-theoretic methods in general.