Zermelo–Fraenkel with Choice

Zermelo–Fraenkel with Choice
Name	Zermelo–Fraenkel with Choice
Other names	ZFC
Type	Axiomatic set theory
Introduced	20th century
Authors	Ernst Zermelo; Abraham Fraenkel; Thoralf Skolem

Zermelo–Fraenkel with Choice

Zermelo–Fraenkel with Choice is a standard axiomatic framework for set theory used in much of contemporary mathematics. It is formulated as a list of axioms intended to capture the behavior of sets while avoiding paradoxes associated with naive set comprehension; its adoption influenced foundations discussions in contexts such as the Hilbert program, the Foundations of Mathematics debates, and interactions with work by figures like David Hilbert, Kurt Gödel, and Paul Cohen. The system underpins theories developed in institutions such as the Institute for Advanced Study, the University of Göttingen, and the Princeton University mathematics departments.

Definition and Axioms

The axioms commonly cited in presentations of Zermelo–Fraenkel with Choice include Extensionality, Pairing, Union, Power Set, Infinity, Replacement, Separation (Schema), Foundation, and the Axiom of Choice; related formulations appear in writings by Ernst Zermelo, Abraham Fraenkel, and Thoralf Skolem. Statements of these axioms were discussed in venues like the Mathematische Zeitschrift and the Proceedings of the Royal Society, and later expounded in textbooks by authors affiliated with Harvard University, University of Chicago, and Massachusetts Institute of Technology. The Axiom of Choice itself has connections to results associated with L. E. J. Brouwer, Henri Lebesgue, Émile Borel, and debates at institutions like the Collège de France. Formal proofs and metamathematical analyses are found in works from Princeton University Press and Cambridge University Press.

Models and Consistency

Models of Zermelo–Fraenkel with Choice are often studied through constructions like von Neumann universes and inner models such as Gödel's constructible universe L; Kurt Gödel proved the relative consistency of Choice and the Generalized Continuum Hypothesis with respect to Zermelo–Fraenkel via the constructible universe, a result presented at the Institute for Advanced Study and published in venues like the Annals of Mathematics. Nonstandard models, ordinal analyses, and models built using techniques from researchers at the University of California, Berkeley, University of Oxford, and Yale University explore properties like well-orderability and definability; individuals such as Dana Scott, Andrey Kolmogorov, and John von Neumann contributed foundational perspectives. Consistency proofs and independence results have been topics at meetings hosted by organizations like the American Mathematical Society and the London Mathematical Society.

Consequences and Equivalent Statements

Adopting Choice yields many classical equivalents and consequences studied by Zermelo, Fraenkel, and later authors: Tychonoff's theorem in topology (connected to work by Andrey Tikhonov), Zorn's lemma (used in algebra by Emmy Noether and Oscar Zariski), well-ordering theorem (discussed by Richard Dedekind), and results in functional analysis traced through contributions by Stefan Banach and Hermann Weyl. Equivalences and relative implications have been analyzed in seminars and monographs from Columbia University, Stanford University, and ETH Zurich, with applications appearing in theorems associated with Élie Cartan, Hassler Whitney, and Jean-Pierre Serre.

Independence Results and Forcing

Independence phenomena were illuminated by Paul Cohen's development of forcing, a technique he announced at conferences like those organized by the Royal Society and published in journals connected to Princeton University Press. Cohen's work established the independence of the Continuum Hypothesis and other statements from Zermelo–Fraenkel with Choice; subsequent refinements by researchers at Yeshiva University, Institute for Advanced Study, and University of Michigan extended forcing and iterated forcing methods. Forcing and independence have influenced work by Saharon Shelah, Donald A. Martin, and John W. Tukey, and have been central topics at meetings of the Association for Symbolic Logic and the Mathematical Association of America.

Variants and Extensions

Variants and extensions of the axiomatic framework include systems that modify or omit the Axiom of Choice, adopt large cardinal axioms, or adjust Foundation; researchers at Princeton University, Paris-Sorbonne University, and University of Bonn have developed inner model theory and large cardinal hierarchies influenced by figures such as Solomon Feferman, W. Hugh Woodin, and William Mitchell. Alternative foundations have been proposed by scholars tied to New York University, University of Cambridge, and Moscow State University, with interactions involving the work of Nikolai Luzin, Andrei Kolmogorov, and Georg Cantor's legacy institutions. Formal systems like second-order formulations and constructive variants relate to efforts at University of Edinburgh and Carnegie Mellon University.

Historical Development and Reception

The historical trajectory began with Georg Cantor's set ideas and continued through formalizations by Ernst Zermelo and improvements by Abraham Fraenkel and Thoralf Skolem; reception varied across schools and eras, debated by figures such as Henri Poincaré, Brouwer, Bertrand Russell, and commentators at institutions like the Royal Society of London and the Académie des Sciences. The mid-20th century consolidation around Zermelo–Fraenkel with Choice was influenced by metamathematical results from Kurt Gödel and Paul Cohen, with further philosophical and technical discussion in journals affiliated with Princeton University and Oxford University Press.

Category:Set theory