Foundations
Foundations of mathematics asks what the discipline itself rests on. Research in this area ranges from the deep technical machinery of mathematical logic — model theory, proof theory, computability, set theory — through category theory and type theory, to the philosophical questions at the interface between mathematics and computer science. Modern foundational work bridges to programming languages (dependent types, proof assistants), homotopy theory (univalent foundations), and computer verification of proofs.
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math.LO Logic
Mathematical logic, model theory, proof theory, set theory, recursion theory, computability.
math.CT Category Theory
Categories, functors, natural transformations, higher categories, (∞,1)-categories, topos theory.
math.HO History and Overview
Historical and expository work on mathematical topics and figures.
Active research areas
The most active problem clusters right now.
Set theory
Large cardinals, inner models, forcing, determinacy axioms, set-theoretic independence results.
Search arXiv →Model theory
Stability theory, o-minimality, connections to algebra and algebraic geometry via Zilber-style trichotomies.
Search arXiv →Proof theory
Ordinal analysis, reverse mathematics, proof complexity, bounded arithmetic.
Search arXiv →Type theory
Dependent types, homotopy type theory (HoTT), cubical type theory, proof assistants (Coq, Lean, Agda).
Search arXiv →Category theory
∞-categories, higher topos theory, applied category theory.
Search arXiv →Landmark results
Major results shaping the field.
- 1931
Gödel’s incompleteness theorems
Formal limits on axiomatic systems containing arithmetic.
- 1963
Cohen’s forcing
Independence of the Continuum Hypothesis from ZFC.
- 2006–2017
Voevodsky’s univalent foundations
Identification of equivalence with identity in dependent type theory; birth of HoTT.
- 2012–present
Mochizuki’s IUTT
Contested proof of the ABC conjecture using Inter-universal Teichmüller theory.
Leading journals
Where current research in this area is published.
- Journal of Symbolic Logic
- Annals of Pure and Applied Logic
- Theoretical Computer Science — Logic in CS
- Logical Methods in Computer Science