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Frege's unofficial arithmetic

Published online by Cambridge University Press:  12 March 2014

Agustín Rayo
Agustín Rayo*
Affiliation:
Department of Logic and Metaphysics, University of St Andrews, St Andrews, KY16 9AL, UK, E-mail: ar29@st-andrews.ac.uk
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Abstract

I show that any sentence of nth-order (pure or applied) arithmetic can be expressed with no loss of compositionality as a second-order sentence containing no arithmetical vocabulary, and use this result to prove a completeness theorem for applied arithmetic. More specifically. I set forth an enriched second-order language L. a sentence of L (which is true on the intended interpretation of L), and a compositionally recursive transformation Tr defined on formulas of L, and show that they have the following two properties: (a) in a universe with at least ℶn−2 objects, any formula of nth-order (pure or applied) arithmetic can be expressed as a formula of L, and (b) for any sentence of is a second-order sentence containing no arithmetical vocabulary, and

Keywords

Second-order logicarithmeticlogicismnominalismFregeHodesBoolos
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 4 , December 2002 , pp. 1623 - 1638
Copyright
Copyright © Association for Symbolic Logic 2002

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