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WHAT RUSSELL SHOULD HAVE SAID TO BURALI–FORTI

  • SALVATORE FLORIO (a1) and GRAHAM LEACH-KROUSE (a2)
Abstract
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.

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*DEPARTMENT OF PHILOSOPHY UNIVERSITY OF BIRMINGHAM BIRMINGHAM, UK E-mail: s.florio@bham.ac.uk
DEPARTMENT OF PHILOSOPHY KANSAS STATE UNIVERSITY MANHATTAN, KS, USA E-mail: gleachkr@ksu.edu
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C Burali-Forti . (1897). Una questione sui numeri transfiniti. Rendiconti del circolo matematico di Palermo, 11, 154164.

J Burgess . (2004). E pluribus unum: Plural logic and set theory. Philosophia Mathematica, 12, 193221.

R. T Cook . (2003). Iteration one more time. Notre Dame Journal of Formal Logic, 44, 6392.

F Ferreira . (2005). Amending Frege’s Grundgesetze der Arithmetik. Synthese, 147, 319.

F. Ferreira & K. F Wehmeier . (2002). On the consistency of the ${\rm{\Delta }}_1^1 $ -CA fragment of Frege’s Grundgesetze. Journal of Philosophical Logic, 31, 301311.

M Glanzberg . (2004). Quantification and realism. Philosophy and Phenomenological Research, 69, 541571.

A. P Hazen . (1986). Logical objects and the paradox of Burali-Forti. Erkenntnis, 24, 283291.

R Heck . (1996). The consistency of predicative fragments of Frege’s Grundgesetze der Arithmetik. History and Philosophy of Logic, 17, 209220.

G Hellman . (2011). On the significance of the Burali-Forti paradox. Analysis, 71, 631637.

H Hodes . (1986). Logicism and the ontological commitments of arithmetic. Journal of Philosophy, 81, 123149.

P. E. B Jourdain . (1904). On the transfinite cardinal numbers of well-ordered aggregates. Philosophical Magazine, 7, 6175.

G. W Leibniz . (1989). Philosophical Papers and Letters. Dordrecht: Kluwer Academic Publishers.

Ø Linnebo . (2004). Predicative fragments of Frege Arithmetic. Bulletin of Symbolic Logic, 10, 153174.

Ø Linnebo . (2010). Pluralities and sets. Journal of Philosophy, 107, 144164.

Ø. Linnebo & R Pettigrew . (2014). Two types of abstraction for structuralism. Philosophical Quarterly, 64, 267283.

P Mancosu . (2015). Grundlagen, section 64: Freges discussion of definitions by abstraction in historical context. History and Philosophy of Logic, 36, 6289.

G. H Moore . & A Garciadiego . (1981). Burali-Forti’s paradox: A reappraisal of its origins. Historia Mathematica, 8, 319350.

C Parsons . (1974a). The liar paradox. Journal of Philosophical Logic, 3, 381412.

C Parsons . (1974b). Sets and classes. Noûs, 8, 112.

H Poincaré . (1912). The latest efforts of the logisticians. The Monist, 22, 524539.

B Russell . (1908). Mathematical logic as based on a theory of types. American Journal of Mathematics, 30, 222262.

S Shapiro . (2003). All sets great and small: And I do mean ALL. Philosophical Perspectives, 17, 467490.

S. Shapiro & A Weir . (1999). New V, ZF and abstraction. Philosophia Mathematica, 7, 293321.

S Simpson . (2009). Subsystems of Second Order Arithmetic. Cambridge, UK: Cambridge University Press.

J. P Studd . (2016). Abstraction reconceived. British Journal for the Philosophy of Science, 67, 579615.

G Uzquiano . (2003). Plural quantification and classes. Philosophia Mathematica, 11, 6781.

S. Walsh & S Ebels-Duggan . (2015). Relative categoricity and abstraction principles. Review of Symbolic Logic, 8, 572606.

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The Review of Symbolic Logic
  • ISSN: 1755-0203
  • EISSN: 1755-0211
  • URL: /core/journals/review-of-symbolic-logic
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