The Consistency of Arithmetic

Abstract

In 2010, Vladimir Voevodsky gave a lecture on "What If Current Foundations of Mathematics Are Inconsistent?" Among other things he said that he was seriously suspicious that an inconsistency in PA (first-order Peano arithmetic) might someday be found. About a year later, Edward Nelson announced that he had discovered an inconsistency not just in PA, but in a small fragment of primitive recursive arithmetic. Soon, Daniel Tausk and Terence Tao independently found a fatal error, and Nelson withdrew his claim, stating that consistency of PA was an "open problem." Many mathematicians may find such claims bewildering. Is the consistency of PA really an open problem? If so, would the discovery of an inconsistency in PA cause all of mathematics to come crashing down like a house of cards? This expository article attempts to address these questions, by sketching and discussing existing proofs of the consistency of PA (including Gentzen's proof and Friedman's relative consistency proof that appeals to the Bolzano-Weierstrass theorem). Since Nelson was a self-avowed formalist, the article also examines the implications of formalism.