Martin's Axiom

Martin's Axiom
Name	Martin's Axiom
Field	Set theory
Introduced	1970
Introduced by	Donald A. Martin, Robert M. Solovay
Related	Continuum hypothesis, Forcing (mathematics), Axiom of Choice

Martin's Axiom is an axiom schema in set theory asserting the existence of filters meeting many dense sets in certain partial orders, formulated to control combinatorial behavior below the size of the power set of the integers. It was introduced to study the interaction of forcing techniques with classical problems such as the Continuum hypothesis and has played a role alongside efforts by Kurt Gödel, Paul Cohen, John von Neumann, and Ernst Zermelo in the development of modern axiomatic frameworks. The axiom is independent of Zermelo–Fraenkel set theory with the Axiom of Choice and has influenced work by researchers including Saharon Shelah, Kenneth Kunen, Jakob B. B. Tarski, and Thomas Jech.

History

Martin's Axiom originated in the collaboration of Donald A. Martin and Robert M. Solovay in the early 1970s as part of an effort to capture forcing consequences weaker than the negation of the Continuum hypothesis. Early antecedents include methods of Paul Cohen's forcing and absoluteness techniques developed by Kurt Gödel and researchers at institutions such as Princeton University and University of California, Berkeley. Developments by Saharon Shelah and Kenneth Kunen expanded the landscape, connecting the axiom to combinatorial principles investigated at Massachusetts Institute of Technology and Harvard University. Subsequent work at venues including Institute for Advanced Study and Mathematical Sciences Research Institute linked the axiom to partition relations studied by Paul Erdős and András Hajnal and to descriptive-set-theoretic questions pursued by Yiannis N. Moschovakis.

Statement

In its usual formulation Martin's Axiom asserts that for any ccc partial order P and any family D of dense subsets of P with |D| < 2^{\aleph_0}, there exists a filter meeting every member of D. The schema quantifies over cardinals below the continuum studied in contexts influenced by the Continuum hypothesis, the notion of countable chain condition as used by Richard Rado and Felix Hausdorff, and ideas from Axiom of Choice considerations associated to Ernest Zermelo. The formulation is often parameterized by a cardinal κ, leading to statements considered by researchers such as Donald A. Martin and Robert M. Solovay that are tailored to combinatorial problems investigated by Paul Cohen and Saharon Shelah at universities including Hebrew University of Jerusalem and Rutgers University.

Consequences and Applications

Under Martin's Axiom many classical statements independent of Zermelo–Fraenkel set theory acquire determinate truth values in models where the axiom holds, paralleling results by Paul Cohen concerning the Continuum hypothesis. It implies that certain topological spaces investigated by Mary Ellen Rudin and Leonard Gillman lack pathological examples such as Dowker spaces under specific cardinal hypotheses, and affects compactness questions studied at University of California, Los Angeles and University of Cambridge. In measure theory and real analysis themes pursued by John von Neumann and Andrey Kolmogorov, Martin's Axiom yields regularity properties for sets of reals linked to work by Solomon Feferman and Akihiro Kanamori. In combinatorial set theory, results related to partition calculus by Paul Erdős and chain conditions connected to Claude Shannon-era information considerations have been clarified by applications of Martin's Axiom, and it interacts with cardinal characteristics of the continuum examined by Hechler and Blass.

Consistency and Independence

Martin's Axiom is consistent relative to Zermelo–Fraenkel set theory with the Axiom of Choice provided the consistency of an inaccessible cardinal or other large-cardinal assumptions employed in the work of Kurt Gödel and Paul Cohen on independence. Cohen forcing and iterated forcing techniques developed by Bob Solovay and Saharon Shelah produce models satisfying Martin's Axiom together with prescribed values of the continuum, mirroring independence proofs central to the research agendas of Harvard University and Princeton University. Conversely, negations of instances of the axiom can be forced by constructions inspired by Paul Cohen's original method, echoing independence phenomena that motivated work by John Conway and Dana Scott.

Variants and Strengthenings

Several strengthenings of Martin's Axiom have been proposed, including bounded forms parameterized by cardinals κ and stronger principles such as the Proper Forcing Axiom studied by Saharon Shelah and Ilijas Farah, and forcing axioms with large-cardinal strength related to research at University of Oxford and University of Bonn. Variants interact with descriptive-set-theoretic regularity properties analyzed by Yiannis N. Moschovakis and with determinacy principles pursued by Donald A. Martin himself and John Steel. Strengthenings such as Martin's Maximum, developed in work connected to Magidor, Foreman, and Matthew Foreman, push the implications toward consequences for the structure of Hereditarily countable sets and for cardinal characteristics studied by Blass and Shelah.

Related Forcing Notions and Models

The ccc forcing notions central to Martin's Axiom are related to partial orders used in Cohen's method and to notions like random forcing and amoeba forcing explored by Robert M. Solovay and Jech; these are studied alongside proper forcing techniques advanced by Saharon Shelah and Stevo Todorčević. Iterated forcing constructions yielding models of Martin's Axiom employ bookkeeping devices refined at research centers such as Institute for Advanced Study and Mathematical Sciences Research Institute, and connect to models where the Continuum hypothesis fails in manners investigated by Paul Cohen and Kurt Gödel. The landscape of models includes those built via collapse forcings and preservation theorems attributed to work at Rutgers University and Hebrew University of Jerusalem, linking to applications in topology, measure theory, and combinatorics by scholars such as Mary Ellen Rudin, Paul Erdős, and Saharon Shelah.

Category:Set theory