Abstract

An idea attributable to Russell serves to extend Zermelo's theory of systems of infinitely long propositions to infinitary relations. Specifically, relations over a given domain $${\mathfrak{D}}$$D of individuals will now be identified with propositions over an auxiliary domain $${\mathfrak{D}^{\mathord{\ast}}}$$D* subsuming $${\mathfrak{D}}$$D. Three applications of the resulting theory of infinitary relations are presented. First, it is used to reconstruct Zermelo's original theory of urelements and sets in a manner that achieves most, if not all, of his early aims. Second, the new account of infinitary relations makes possible a concise characterization of parametric definability with respect to a purely relational structure. Finally, based on his foundational philosophy of the primacy of the infinite, Zermelo rejected Godel's First Incompleteness Theorem; it is shown that the new theory of infinitary relations can be brought to bear, positively, in that connection as well.

Highlights

  • Zermelo is widely recognized for having introduced, early in his career, persuasive axioms describing the realm of urelements and sets that had been investigated by Cantor and Dedekind

  • In contrast, is Zermelo’s theory of infinitely long propositions, the focus of three papers published at the end of his career

  • Zermelo’s theory of infinitely long propositions is variable-free, and we follow him, and Russell, in this regard as we extend that theory to relations

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Summary

Introduction

Zermelo is widely recognized for having introduced, early in his career, persuasive axioms describing the realm of urelements and sets that had been investigated by Cantor and Dedekind. In contrast, is Zermelo’s theory of infinitely long propositions, the focus of three papers published at the end of his career. The latter theory, like the axiomatization, was to make a difference for the foundations of mathematics, as Zermelo saw things. Substance, from Russell’s substitutional theory as we proceed to extend Zermelo’s theory of infinitely long propositions, described, to a theory of infinitary relations, set out in Sect. Zermelo’s theory of infinitely long propositions is variable-free, and we follow him, and Russell, in this regard as we extend that theory to relations. Even a fervent believer in variables may ask, Can a theory of relations manage without them? As Russell showed, the answer is yes: Variables themselves are not essential, whereas some notion of substitution is key

Zermelo’s Systems of Infinitely Long Propositions
A Substitutional Theory of Infinitary Relations
Zermelo Set Theory Reconstructed
Definability
Arithmetic Expressibility and Infinitary Relations
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