Notre Dame Journal of Formal Logic

A Single Axiom for Set Theory

David Bennett

Abstract

Axioms in set theory typically have the form $\forall z \exists y\forall x(x \in y \leftrightarrow F x z )$, where $F$ is a relation which links $x$ with $z$ in some way. In this paper we introduce a particular linkage relation $L$ and a single axiom based on $L$ from which all the axioms of $\mathrm{Z}$ (Zermelo set theory) can be derived as theorems. The single axiom is presented both in informal and formal versions. This calls for some discussion of pertinent features of formal and informal axiomatic method and some discussion of pertinent features of the system $\mathrm{S}$ of set theory to be erected on the single axiom. $\mathrm{S}$ is shown to be somewhat stronger than $\mathrm{Z}$, but much weaker than $\mathrm{ZF}$ (Zermelo-Fraenkel set theory).

Article information

Source
Notre Dame J. Formal Logic, Volume 41, Number 2 (2000), 152-170.

Dates
First available in Project Euclid: 25 November 2002

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1038234609

Digital Object Identifier
doi:10.1305/ndjfl/1038234609

Mathematical Reviews number (MathSciNet)
MR1932227

Zentralblatt MATH identifier
1015.03050

Subjects
Primary: 03E30: Axiomatics of classical set theory and its fragments
Secondary: 00A30: Philosophy of mathematics [See also 03A05]

Keywords
set Cantorian axiom links descendant single foundation basic

Citation

Bennett, David. A Single Axiom for Set Theory. Notre Dame J. Formal Logic 41 (2000), no. 2, 152--170. doi:10.1305/ndjfl/1038234609. https://projecteuclid.org/euclid.ndjfl/1038234609


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References

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