Review of Symbolic Logic 10 (4):682-718 (2017)

Graham Leach-Krouse
Kansas State University
Salvatore Florio
University of Birmingham
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.
Keywords paradox  Burali-Forti  ordinals  abstraction  logicism  higher-order logic  Russell  no class theory
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DOI 10.1017/S1755020316000484
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References found in this work BETA

The Principles of Mathematics.Bertrand Russell - 1903 - Revue de Métaphysique et de Morale 11 (4):11-12.
Pluralities and Sets.Øystein Linnebo - 2010 - Journal of Philosophy 107 (3):144-164.
Logicism and the Ontological Commitments of Arithmetic.Harold T. Hodes - 1984 - Journal of Philosophy 81 (3):123-149.
The Liar Paradox.Charles Parsons - 1974 - Journal of Philosophical Logic 3 (4):381 - 412.
On Some Difficulties in the Theory of Transfinite Numbers and Order Types.Bertrand Russell - 1905 - Proceedings of the London Mathematical Society 4 (14):29-53.

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Citations of this work BETA

Bolzano’s Mathematical Infinite.Anna Bellomo & Guillaume Massas - forthcoming - Review of Symbolic Logic:1-80.
Classes, Why and How.Thomas Schindler - 2019 - Philosophical Studies 176 (2):407-435.

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