Gödel–Bernays set theory

Gödel–Bernays set theory
Name	Gödel–Bernays set theory
Author	Kurt Gödel; Paul Bernays
Introduced	1920s–1930s
Language	German
Genre	Axiomatic set theory

Gödel–Bernays set theory is an axiomatic theory formulated to formalize a two-tier ontology of set theory with distinct sorts for sets and classes and to analyze foundational relations to Zermelo–Fraenkel set theory and consistency proofs associated with work by Kurt Gödel and Paul Bernays. It has been used in metamathematical investigations related to independence phenomena explored by John von Neumann, Ernst Zermelo, Abraham Fraenkel, and later refinements informing research of Dana Scott, Solomon Feferman, and Paul Cohen. The theory plays a role in discussions involving models developed by Vladimir Voevodsky, proof-theoretic studies by Gerhard Gentzen, and applications considered in seminars at Institute for Advanced Study and Princeton University.

Overview and Motivation

Gödel–Bernays set theory was motivated by questions raised in seminars of Hilbert and correspondence among David Hilbert, Emmy Noether, and Kurt Gödel concerning formalization of classes such as the class of all ordinals and classes used in work of Ernst Zermelo and John von Neumann. The formulation distinguishes classes from sets to avoid paradoxes like those addressed by Bertrand Russell in correspondence with Alfred North Whitehead and to align with cumulative-hierarchy intuitions examined in lectures at University of Göttingen and ETH Zurich. The motivation connects to model constructions later employed by Kurt Gödel in his constructible universe L and to independence techniques pioneered by Paul Cohen at Harvard University and Stanford University.

Formal Language and Axioms

The formal language uses two sorts, paralleling approaches of Alonzo Church and syntactic frameworks from David Hilbert's school, with membership and class comprehension schemas analogous to axioms studied by Ernst Zermelo, Abraham Fraenkel, and formalized in seminar notes of Paul Bernays. Axioms include extensionality, class comprehension for formulas without quantification over classes, pair, union, power set relative to sets as formulated in correspondence between Kurt Gödel and Paul Bernays, and a global choice principle related to choice results investigated by Ernst Zermelo and later by John von Neumann. The schema structure reflects metamathematical concerns also treated by Gerhard Gentzen in proof consistency analyses and by Dana Scott in model-theoretic presentations.

Models and Relative Consistency

Models are typically constructed by interpreting classes as definable collections over Zermelo–Fraenkel set theory models or by adopting the two-sorted universes of John von Neumann used in early set-theoretic research at Institute for Advanced Study. Relative consistency proofs relate the theory to Zermelo–Fraenkel set theory with the Axiom of Choice as examined by Kurt Gödel and independence techniques of Paul Cohen; consistency transfers employ inner models like Gödel’s constructible universe L and forcing arguments developed at Harvard University. Model-theoretic analyses leverage methods from Alfred Tarski and Dana Scott and have implications studied in seminars at University of California, Berkeley and Princeton University.

Relationship to Zermelo–Fraenkel Set Theory

The relationship to Zermelo–Fraenkel set theory is close: many results show conservativity over ZF for statements about sets as demonstrated in expositions by Paul Bernays and later by Solomon Feferman. Theories differ in ontology but agree on theorems expressible in the language of sets, a correspondence explored in writings by Kurt Gödel, John von Neumann, and surveys at Institut des Hautes Études Scientifiques. Connections to the Axiom of Choice recall debates involving Ernst Zermelo and influence work on large cardinals by Kurt Gödel and researchers at University of California, Berkeley.

Class and Set Constructions

Class formation rules permit classes defined by formulas without quantification over classes, echoing comprehension restrictions considered by Bertrand Russell and formal treatments by Paul Bernays and Kurt Gödel. Concrete constructions include ordinals and cardinals in the style of Ernst Zermelo and the von Neumann ordinal assignment used by John von Neumann, with cumulative-hierarchy perspectives discussed at University of Göttingen and in notes circulated to scholars at Princeton University. The interplay of sets and proper classes appears in literature by Dana Scott and Solomon Feferman and underpins constructions of inner models and definable classes used in large-cardinal investigations.

Variants and Extensions

Variants include formulations equivalent to the original in strength and conservative over Zermelo–Fraenkel set theory for set statements, as developed in alternative presentations by Paul Bernays and later expositors such as Solomon Feferman and Thomas Jech. Extensions incorporate global choice or additional comprehension schemes influenced by work of John von Neumann and investigations at Institute for Advanced Study into class forcing and large-cardinal axioms, with modern treatments appearing in courses at Harvard University and publications associated with American Mathematical Society conferences.

Historical Development and Attribution

The theory traces to collaborative developments during the 1920s and 1930s among Kurt Gödel, Paul Bernays, John von Neumann, and commentators including Ernst Zermelo and Abraham Fraenkel, with archival material from University of Göttingen and correspondence preserved at Institute for Advanced Study. Attribution has been discussed in historical studies referencing lectures and notes circulated at Princeton University and in historiography by scholars linked to Institute for Advanced Study and ETH Zurich.

Category:Axiomatic set theory