A Theory of Infinitary Relations Extending Zermelo's Theory of Infinitary Propositions

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• 作者：R. Gregory Taylor
• 关键词：Definable set ; Incompleteness Theorem for Peano Arithmetic ; Infinitary relation ; Russell’s substitutional theory ; Zermelo set theory
• 刊名：Studia Logica
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：104
• 期：2
• 页码：277-304
• 全文大小：636 KB
• 刊物类别：Humanities, Social Sciences and Law
• 刊物主题：Philosophy
  Logic
  Mathematical Logic and Foundations
  Computational Linguistics
  
• 出版者：Springer Netherlands
• ISSN：1572-8730

文摘

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 \({\mathfrak{D}}\) of individuals will now be identified with propositions over an auxiliary domain \({\mathfrak{D}^{\mathord{\ast}}}\) subsuming \({\mathfrak{D}}\). 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 Gödel’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.