Zermelo–Fraenkel set theory

Zermelo–Fraenkel set theory
Name	Zermelo–Fraenkel set theory
Field	Mathematical logic
Introduced	20th century
Notable people	Ernst Zermelo, Abraham Fraenkel, Thoralf Skolem, John von Neumann, Kurt Gödel

Contents

Zermelo–Fraenkel set theory is the standard axiomatic foundation for much of modern Mathematics, used to formalize concepts in Peano arithmetic, Real analysis, Algebraic topology, Category theory, and Measure theory. Developed in dialogue with work by Ernst Zermelo, Abraham Fraenkel, Thoralf Skolem, and influenced by results of Kurt Gödel and Paul Cohen, it underpins formal investigations in Hilbert's problems, Gödel's incompleteness theorems, Continuum hypothesis, and constructions in Functional analysis.

Introduction

Zermelo–Fraenkel set theory presents an axiomatic system intended to avoid paradoxes identified by Bertrand Russell, Gottlob Frege, and critiques from Richard Dedekind, while enabling formal development used by David Hilbert, Emil Artin, Andrey Kolmogorov, Henri Lebesgue, and Émile Borel. The axioms were shaped through contributions and critiques from Ernst Zermelo, Abraham Fraenkel, and Thoralf Skolem, and later studied by John von Neumann, Wacław Sierpiński, Paul Cohen, and Kurt Gödel. The framework serves as the lingua franca for researchers at institutions such as Princeton University, University of Göttingen, University of Cambridge, Harvard University, and University of Chicago.

The axioms include Extensionality, Pairing, Union, Power Set, Infinity, Foundation (Regularity), Replacement, and Separation, which were refined by Ernst Zermelo and Abraham Fraenkel and analyzed by Thoralf Skolem, Kurt Gödel, Paul Cohen, and John von Neumann. Each axiom interacts with constructions used by Georg Cantor, Richard Dedekind, Leopold Kronecker, Évariste Galois, and Augustin-Louis Cauchy in formalizing number systems and sets employed in Algebraic geometry, Differential geometry, Set-theoretic topology, and Probability theory. The Axiom of Replacement, emphasized by Abraham Fraenkel and formalized in systems used at Stanford University and Massachusetts Institute of Technology, enables transfinite recursive definitions akin to methods in Ernst Mach and procedures considered by Henri Poincaré. The Axiom of Choice, treated as an independent principle by Kurt Gödel and Paul Cohen, connects to results like Zorn's Lemma used by Samuel Eilenberg and Norman Steenrod in homological algebra and by Alfred Tarski in fixed-point theorems.

Model-theoretic and proof-theoretic analysis of Zermelo–Fraenkel systems has been driven by work of Kurt Gödel on relative consistency, Paul Cohen on forcing, and model constructions by Thoralf Skolem and Alonzo Church; these studies intersect with methods used in Model theory by Saharon Shelah and Wilfrid Hodges at locations like University of Oxford and University of Cambridge. Gödel's constructible universe L, developed by Kurt Gödel and elaborated by J. L. Kelley and Dana Scott, provides inner models that show relative consistency of certain statements, while forcing techniques invented by Paul Cohen produced independence results for the Continuum hypothesis and the Axiom of Choice, paralleling independence phenomena studied by Alfred Tarski and Gerhard Gentzen. Large cardinal axioms introduced by Kurt Gödel and advanced by William Reinhardt, Azriel Levy, Solomon Feferman, and W. Hugh Woodin extend consistency considerations into research programs at institutions such as Institute for Advanced Study and Princeton University.

Common variants include ZF (Zermelo–Fraenkel without the Axiom of Choice), ZFC (ZF plus the Axiom of Choice), and extensions with large cardinals or determinacy axioms considered by Donald Martin, John Myhill, Thomas Jech, and Akihiro Kanamori. Alternative foundations related to ZF/ZFC have been proposed or analyzed by John von Neumann (von Neumann–Bernays–Gödel set theory), Ernst Zermelo (original Zermelo set theory), and Hugh Woodin (axioms for determinacy), and are compared with approaches from Category theory advocated by Saunders Mac Lane and Samuel Eilenberg at Columbia University, Yale University, and University of Chicago. Constructive and predicative variants interact with work by Luitzen Brouwer, Per Martin-Löf, Arend Heyting, and Hermann Weyl in intuitionism and type theory research at Uppsala University and Göteborg University.

Zermelo–Fraenkel axioms underpin mainstream proofs and constructions found in texts by Paul Halmos, Walter Rudin, Serge Lang, H. S. M. Coxeter, and John Conway, influencing curricula at University of Oxford, University of Cambridge, Harvard University, Princeton University, and Yale University. They shape results in Set-theoretic topology used by Ryszard Engelking and Mary Ellen Rudin, inform cardinal arithmetic studied by Georg Cantor and Easton, and support measure-theoretic foundations applied by Andrey Kolmogorov and Paul Lévy. Independence and model techniques originating with Kurt Gödel and Paul Cohen continue to guide research programs by Saharon Shelah, W. Hugh Woodin, and Thomas Jech on problems such as the Continuum hypothesis and the structure of the Set-theoretic universe.