Kelley–Morse set theory

Kelley–Morse set theory
Name	Kelley–Morse set theory
Other names	KM
Subject	Set theory
Introduced	1955
Contributors	Paul R. Halmos; John L. Kelley; Anthony Morse
Type	Axiomatic set theory

Kelley–Morse set theory is an axiomatic set theory designed to formalize classes and sets with strong comprehension principles, extending Zermelo–Fraenkel set theory and complementing work in von Neumann–Bernays–Gödel set theory. It provides a framework for discussing proper classes in a two-sorted language and supports powerful principles used in research following developments by John von Neumann, Paul Cohen, and Kurt Gödel. The theory has been influential in investigations related to large cardinals, inner model theory, and the metamathematics of category theory, analysis, and forcing.

Overview

Kelley–Morse set theory presents a two-sorted system distinguishing sets and classes, inspired by ideas of John von Neumann and formal approaches advanced by Alonzo Church and Stephen Kleene. It augments the expressive resources used in Zermelo set theory and Zermelo–Fraenkel set theory by allowing class comprehension for formulas with quantification over sets and classes, connecting to methods from Kurt Gödel’s constructible universe and techniques developed by Paul Cohen in independence proofs. Researchers in Harvard University, Princeton University, University of California, Berkeley, and University of Cambridge have used the theory to study interactions with hypotheses like the Axiom of Choice and large cardinal axioms formulated by figures such as Solomon Feferman and William Reinhardt.

Axioms and formal language

The formal language of Kelley–Morse uses two sorts governed by axioms resembling those in von Neumann–Bernays–Gödel set theory and axioms reflecting work from Alfred Tarski and Thoralf Skolem. Basic axioms assert extensionality and pairing similar to formulations by Ernst Zermelo and Abraham Fraenkel, along with comprehension schemes permitting class comprehension for formulas with quantifiers over sets and classes, paralleling schema considerations raised by Georg Cantor and elaborated by Paul Bernays. Replacement and foundation principles echo treatments found in writings by Kurt Gödel and John Conway. The global choice principle often discussed in relation to Kelley–Morse was debated by scholars including Kurt Gödel and Paul Cohen.

Models and consistency strength

Models of Kelley–Morse have been studied via constructions akin to Gödel’s constructible universe L and through forcing techniques pioneered by Paul Cohen and refined by Azriel Lévy and Robert Solovay. Relative consistency results relate Kelley–Morse to strong large cardinal hypotheses studied by Kurt Gödel, Felix Hausdorff, and Gerald Sacks, while inner model theory work by Donald A. Martin and Leo Harrington investigates determinacy principles and their implications inside KM-like frameworks. Consistency strength comparisons involve research by Solomon Feferman, William Tait, and Kenneth Kunen, and interact with the theory of measurable, supercompact, and huge cardinals associated with John Steel and W. Hugh Woodin.

Relations to other set theories

Kelley–Morse can be compared to von Neumann–Bernays–Gödel set theory and to the first-order axiomatizations explored by Zermelo and Fraenkel. Connections to New Foundations and to constructive approaches linked to L. E. J. Brouwer and Arend Heyting appear in comparative studies by Dana Scott and Per Martin-Löf. The role of classes in KM resonates with categorical foundations considered by Saunders Mac Lane and William Lawvere, while metatheoretic properties have been explored in work by Georg Kreisel and Stephen Cole Kleene.

Applications and consequences

Kelley–Morse is used in analyses of class forcing as developed by James E. Baumgartner and Sy Friedman, and in formulations of global choice relevant to work by Paul Cohen and Kenneth Kunen. Its expressive power has been applied in descriptive set theory research connected to Donald A. Martin and Yiannis N. Moschovakis, and in algebraic set theory dialogues involving Michael Makkai and Joyal’s categorical collaborators. Philosophical and foundational consequences have been debated in venues associated with Harvard University, Princeton University, and conferences honoring Kurt Gödel and Alfred Tarski.

History and development

Kelley–Morse emerged from mid-20th-century efforts by researchers influenced by John von Neumann, Paul R. Halmos, John L. Kelley, and Anthony Morse. Subsequent refinements and analyses were contributed by logicians at institutions including Massachusetts Institute of Technology, University of California, Berkeley, University of Oxford, and Hebrew University of Jerusalem. The development parallels milestones in the work of Kurt Gödel on constructibility, Paul Cohen on forcing, and later large cardinal investigations by Solomon Feferman, Kenneth Kunen, and W. Hugh Woodin. Ongoing research communities at Institute for Advanced Study and international symposia continue to explore KM’s role alongside evolving approaches in set-theoretic geology and inner model theory championed by Jech and Steel.

Category:Set theory