A single axiom for set theory. (English) Zbl 1015.03050

Bennett, David

Notre Dame J. Formal Logic 41, No. 2, 152-170 (2000).

Summary: Axioms in set theory typically have the form \(\forall z\exists y\forall x(x\in y\leftrightarrow Fxz)\), 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 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 S of set theory to be erected on the single axiom. S is shown to be somewhat stronger than Z, but much weaker than ZF (Zermelo-Fraenkel set theory).

MSC:

03E30	Axiomatics of classical set theory and its fragments
00A30	Philosophy of mathematics

Keywords:

Zermelo set theory; linkage relation; single axiom

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