Von Neumann–Bernays–Gödel set theory

Von Neumann–Bernays–Gödel set theory
Name	Von Neumann–Bernays–Gödel set theory
Discipline	Mathematical logic
Introduced	1920s–1930s
Notable figures	John von Neumann; Paul Bernays; Kurt Gödel

Contents

Von Neumann–Bernays–Gödel set theory is a formal axiomatic system for set theory that distinguishes between sets and classes and provides an alternative foundational framework to axioms used in mainstream twentieth-century logic. It is used in foundational work associated with figures and institutions such as John von Neumann, Paul Bernays, Kurt Gödel, Princeton University, Institute for Advanced Study, and texts linked to David Hilbert, Emil Post, and Alonzo Church. The theory is relevant to debates among proponents around Zermelo–Fraenkel set theory, Georg Cantor, Bertrand Russell, Ernst Zermelo, and Abraham Fraenkel.

Introduction

Von Neumann–Bernays–Gödel set theory was developed to refine earlier proposals by Ernst Zermelo and Bertrand Russell and to respond to paradoxes discussed by Gottlob Frege and Giuseppe Peano. Influenced by work at the University of Göttingen and correspondence among David Hilbert, Emil Artin, Felix Hausdorff, the system formalizes a two-sorted language enabling discourse about proper classes and sets, engaging with ideas found in writings of John von Neumann and editorial work by Paul Bernays and later commentary by Kurt Gödel. Its formulation is situated in the milieu that included institutions such as the University of Vienna, Princeton University, and ETH Zurich.

The axioms are presented in a language distinguishing variables for sets and variables for classes, echoing techniques used by Ernst Zermelo and refined in methods linked to Abraham Fraenkel and Alonzo Church. Core principles mirror replacement, foundation, and choice themes discussed by Zermelo and debated by Kurt Gödel and Paul Cohen, while employing class comprehension restricted similarly to restrictions in writings by Bertrand Russell and Gottlob Frege. Schemes analogous to the Replacement Schema and the Axiom of Choice are treated in the spirit of analyses by Ernst Zermelo and the independence results obtained by Paul Cohen and later scholars associated with Harvard University and University of California, Berkeley. Formal proofs within the system draw on techniques referenced by Gerhard Gentzen and Kurt Gödel and are taught in seminars influenced by curricula at Princeton University and University of Oxford.

Relative consistency results connect the theory to models constructed in the tradition of Kurt Gödel's constructible universe and methods employed by Paul Cohen involving forcing at institutions like Harvard University and University of Illinois Urbana–Champaign. Model constructions reference approaches developed at Institute for Advanced Study and by researchers affiliated with University of Cambridge and Yale University. Theoretical relationships use inner models, transitive classes, and hierarchies examined in scholarship by Kurt Gödel, W. Hugh Woodin, Dana Scott, and Ronald Jensen, and intersect with independence phenomena explored by Paul Cohen, James E. Baumgartner, and Silver. Consistency proofs often cite correspondences with systems associated with Ernst Zermelo and Abraham Fraenkel and invoke methods familiar to logicians from MIT and University of California, Los Angeles.

Comparisons contrast the two-sorted approach with the single-sorted language of Ernst Zermelo and Abraham Fraenkel as codified in the Zermelo–Fraenkel axioms used at places like Princeton University and Harvard University. The relationship addresses how Replacement, Choice, and Regularity are manifested, building on debates involving Kurt Gödel, Paul Cohen, and commentators from University of Chicago and Columbia University. Researchers such as Solomon Feferman, John Conway, and Azriel Lévy have contributed perspectives comparing expressive power and convenience for formalizing category-theoretic work discussed at Massachusetts Institute of Technology and University of Cambridge.

The theory is applied in formal treatments of category theory, model theory, and proof theory developed in line with programs at Institute for Advanced Study, Princeton University, and University of California, Berkeley. It has been employed by mathematicians and logicians associated with Kurt Gödel, Paul Bernays, and later scholars at Harvard University and Bonn University for clean handling of large collections encountered in work influenced by Alexander Grothendieck, Saunders Mac Lane, and Samuel Eilenberg. Uses include formalizing large-cardinal hypotheses discussed by William Reinhardt, W. Hugh Woodin, and Robert M. Solovay, and in expositions by contributors affiliated with University of Oxford and École Normale Supérieure.

Origins trace to unpublished notes and letters involving John von Neumann at Princeton University and editorial elaborations by Paul Bernays in contexts connected to University of Göttingen and University of Zurich. Subsequent analysis and dissemination were influenced by Kurt Gödel and later commentators from institutions such as Institute for Advanced Study, University of Vienna, and ETH Zurich. The development intersects with the broader history of logic involving figures like David Hilbert, Bertrand Russell, Ernst Zermelo, Alonzo Church, Paul Cohen, and modern researchers at Harvard University and Massachusetts Institute of Technology.