Joel David Hamkins

Mathematician and philosopher specialising in set theory, mathematical logic, and the philosophy of mathematics. Professor of Logic at Oxford. Number one highest-rated user on MathOverflow (research mathematics Q&A). Author of Proof and the Art of Mathematics and Lectures on the Philosophy of Mathematics. Runs the blog infinitelymore.xyz.

Background

Works on the foundations of mathematics, with particular focus on forcing, large cardinals, and the philosophical implications of set-theoretic independence. Known for his mathematical multiverse position: set-theoretic pluralism holds that there is no single true model of set theory — multiple non-isomorphic ZFC universes are equally mathematically real. This contrasts with the unificationist programme that seeks new axioms to settle independent questions like the Continuum Hypothesis.

Known for: the mathematical multiverse / set-theoretic pluralism; the committee and fruit-salad anthropomorphisations of Cantor’s power-set theorem; framing Russell’s paradox as “Russell’s theorem”; unusually clear pedagogical exposition of Gödel, Cantor, and ZFC foundations.

Appearances in this wiki

Key positions

• Infinity has multiple sizes: the real numbers are uncountably infinite — provably larger than the naturals (Cantor’s diagonal argument)
• Diagonalisation is the master proof technique of mathematical logic, underlying Cantor, Russell, Gödel, and Turing
• Gödel’s incompleteness theorems decisively destroy both goals of Hilbert’s programme: no complete consistent arithmetic theory, and no theory proves its own consistency
• Truth is semantic (Tarski: what is the case in a structure); proof is syntactic (symbol manipulation); these come apart — incompleteness shows true statements need not be provable
• Mathematical multiverse: no single privileged model of ZFC; the Continuum Hypothesis is true in some models, false in others, analogous to the parallel postulate across geometries
• ZFC axioms, including the Axiom of Choice, are fundamentally logical in character — logicism is vindicated by set-theoretic foundationalism