What Russell Should Have Said to Burali-Forti

Research output: Contribution to journalArticle

Authors

Colleges, School and Institutes

External organisations

  • Kansas State University

Abstract

The paradox that appears under Burali-Forti’s name in many textbooks of set theory is a clever piece of reasoning leading to an unproblematic theorem. The theorem asserts that the ordinals do not form a set. For such a set would be—absurdly—an ordinal greater than any ordinal in the set of all ordinals. In this article, we argue that the paradox of Burali-Forti is first and foremost a problem about concept formation by abstraction, not about sets. We contend, furthermore, that some hundred years after its discovery the paradox is still without any fully satisfactory resolution. A survey of the current literature reveals one key assumption of the paradox that has gone unquestioned, namely the assumption that ordinals are objects. Taking the lead from Russell’s no class theory, we interpret talk of ordinals as an efficient way of conveying higher-order logical truths. The resulting theory of ordinals is formally adequate to standard intuitions about ordinals, expresses a conception of ordinal number capable of resolving Burali-Forti’s paradox, and offers a novel contribution to the longstanding program of reducing mathematics to higher-order logic.

Details

Original languageEnglish
JournalThe Review of Symbolic Logic
Early online date27 Feb 2017
Publication statusE-pub ahead of print - 27 Feb 2017

Keywords

  • Burali-Forti , Russell , abstraction , higher-order logic , logicism , no class theory , ordinals , paradox

ASJC Scopus subject areas