large cardinals

large cardinals
Name	Large cardinals
Field	Set theory
Introduced	20th century
Notable	Kurt Gödel; Paul Cohen; W. Hugh Woodin; John von Neumann; Andrey Kolmogorov
Related	Axiom of Choice; Zermelo–Fraenkel set theory; Continuum Hypothesis; Forcing

large cardinals

Large cardinals are strong axioms of infinity studied within Zermelo–Fraenkel set theory and its extensions such as Zermelo set theory and theories motivated by work of Kurt Gödel and John von Neumann. They assert existence of infinite cardinal numbers with extraordinary combinatorial, structural, or elementary-embedding properties, influencing questions about the Continuum Hypothesis, the Axiom of Choice, and models constructed by Paul Cohen and subsequent researchers. Research on large cardinals connects to contributions from W. Hugh Woodin, Solomon Feferman, Robert M. Solovay, Dana Scott, and Donald A. Martin.

Introduction

Large cardinal axioms extend Zermelo–Fraenkel set theory (often with the Axiom of Choice) by positing cardinals whose existence implies rich structural features in models developed by Kurt Gödel and investigated through techniques originating with Paul Cohen and Azriel Lévy. Their study interacts with work of Georg Cantor on transfinite numbers and later developments by Ernst Zermelo, Abraham Fraenkel, and Thoralf Skolem. Results by Harvey Friedman and Kenneth Kunen have clarified their role relative to classical problems like the Continuum Hypothesis and statements in descriptive set theory advanced by Yiannis N. Moschovakis.

Definitions and Hierarchy

Definitions of particular large cardinal notions use objects and techniques introduced by John von Neumann, Dana Scott, and Solomon Feferman, including elementary embeddings, ultrafilters, measures, and combinatorial principles. The hierarchy is typically arranged by consistency strength and closure properties studied by James E. Baumgartner and Menachem Magidor. Starting points include measurable, inaccessible, and Mahlo cardinals, while stronger notions involve supercompact, huge, and extendible cardinals analyzed in work by William J. Mitchell and Philip Welch. The formal ordering of strength has been refined using inner model theory developed by John R. Steel and W. Hugh Woodin.

Examples of Large Cardinals

Classical examples trace to early investigations by Ernst Zermelo and Kurt Gödel: inaccessible cardinals, introduced in the context of Zermelo set theory; measurable cardinals, characterized using ultrafilters in work by Solomon Feferman and Dana Scott; and Mahlo cardinals studied by Paul Erdős and András Hajnal. Stronger examples include supercompact cardinals explored by Menachem Magidor and Michael J. Mitchel, huge cardinals considered by Jech and Ken Kunen, and extendible cardinals examined by John R. Steel and W. Hugh Woodin. Other specialized notions like Woodin cardinals were developed by W. Hugh Woodin and studied by Hugh Woodin in connection with determinacy results by Donald A. Martin and John R. Steel.

Consistency and Relative Strength

Relative consistency results trace to Gödel’s constructible universe and Cohen’s forcing technique; milestones include Gödel’s relative consistency of the Continuum Hypothesis and Cohen’s independence proofs involving Paul Cohen. Work by Kurt Gödel, Paul Cohen, Solomon Feferman, and Hugh Woodin uses inner model theory and large cardinals to calibrate consistency strength. Results by Kenneth Kunen and William J. Mitchell established limits such as Kunen’s inconsistency, while comparative analyses by Joel David Hamkins and Victoria Gitman explore forcing extensions and preservation. Meta-mathematical calibrations often cite paradigms from Gerhard Gentzen and later proof-theoretic investigations by Michael Rathjen.

Applications and Consequences in Set Theory

Large cardinals underpin many results across set theory, from determinacy principles advanced by Donald A. Martin and John R. Steel to the structure of the projective hierarchy studied by Yiannis N. Moschovakis and Alexander S. Kechris. They inform inner model constructions like the core model developed by Donald A. Martin collaborators and Mitchell techniques, and influence combinatorial principles used by Paul Erdős and András Hajnal. Connections to descriptive set theory are central in work by Alexander S. Kechris, John R. Steel, and W. Hugh Woodin, while interactions with forcing and large-cardinal indestructibility were advanced by Richard Laver and Menachem Magidor.

Independence Results and Forcing

Forcing, pioneered by Paul Cohen, is the primary method to obtain independence results relative to large cardinal hypotheses; subsequent refinements by Robert M. Solovay, Kenneth Kunen, and Martin Davis clarified preservation and collapse arguments. Independence of statements like the Continuum Hypothesis and combinatorial assertions often uses collapse forcings and iterated forcing developed by John E. Baumgartner and Saharon Shelah. Preservation theorems and indestructibility results owe much to work by Richard Laver, Menachem Magidor, and Joel David Hamkins, while descriptive-set-theoretic independence was influenced by W. Hugh Woodin.

Historical Development and Key Contributors

The study began with foundational work by Georg Cantor and axiomatization by Ernst Zermelo and Abraham Fraenkel, with fundamental contributions from Kurt Gödel on constructibility and Paul Cohen on forcing. Seminal contributors to large cardinal theory include Dana Scott, Solomon Feferman, Kenneth Kunen, Robert M. Solovay, Donald A. Martin, Menachem Magidor, William J. Mitchell, John R. Steel, W. Hugh Woodin, and Joel David Hamkins. Ongoing research continues in centers such as Institute for Advanced Study communities and mathematics departments at institutions associated with these researchers.

Category:Set theory