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The anatytic conception of truth and the foundations of arithmetic

Published online by Cambridge University Press:  12 March 2014

Peter Apostoli
Peter Apostoli*
Affiliation:
Department of Philosophy, University of Toronto, 215 Huron Street, 9th Floor, Toronto, Ontario, CanadaM5S 1A1, E-mail: [email protected]
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Extract

Until very recently, it was thought that there couldn't be any current interest in logicism as a philosophy of mathematics. Indeed, there is an old argument one often finds that logicism is a simple nonstarter just in virtue of the fact that if it were a logical truth that there are infinitely many natural numbers, then this would be in conflict with the existence of finite models. It is certainly true that from the perspective of model theory, arithmetic cannot be a part of logic. However, it is equally true that model theory's reliance on a background of axiomatic set theory renders it unable to match Frege's Theorem, the derivation within second order logic of the infinity of the number series from the contextual “definition” of the cardinality operator. Called “Hume's Principle” by Boolos, the contextual definition of the cardinality operator is presented in Section 63 of Grundlagen, as the statement that, for any concepts F and G,

the number of F s = the number of G s

if, and only if,

F is equinumerous with G.

The philosophical interest in Frege's Theorem derives from the thesis, defended for example by Crispin Wright, that Hume's principle expresses our pre-analytic conception of assertions of numerical identity. However, Boolos cites the very fact that Hume's principle has only infinite models as grounds for denying that it is logically true: For Boolos, Hume's principle is simply a disguised axiom of infinity.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 1 , March 2000 , pp. 33 - 102
Copyright
Copyright © Association for Symbolic Logic 2000

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