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THE STRENGTH OF ABSTRACTION WITH PREDICATIVE COMPREHENSION

Published online by Cambridge University Press:  29 March 2016

SEAN WALSH
SEAN WALSH*
Affiliation:
DEPARTMENT OF LOGIC AND PHILOSOPHY OF SCIENCE 5100 SOCIAL SCIENCE PLAZA UNIVERSITY OF CALIFORNIA, IRVINE IRVINE, CA 92697-5100, USAE-mail: swalsh108@gmail.com or walsh108@uci.eduURL: http://www.walsh108.org
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Abstract

Frege’s theorem says that second-order Peano arithmetic is interpretable in Hume’s Principle and full impredicative comprehension. Hume’s Principle is one example of an abstraction principle, while another paradigmatic example is Basic Law V from Frege’s Grundgesetze. In this paper we study the strength of abstraction principles in the presence of predicative restrictions on the comprehension schema, and in particular we study a predicative Fregean theory which contains all the abstraction principles whose underlying equivalence relations can be proven to be equivalence relations in a weak background second-order logic. We show that this predicative Fregean theory interprets second-order Peano arithmetic (cf. Theorem 3.2).

Keywords

abstraction principlespredicativitysecond-order arithmeticFregeFrege’s Theorem

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Type
Articles
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Bulletin of Symbolic Logic , Volume 22 , Issue 1 , March 2016 , pp. 105 - 120
Copyright
Copyright © The Association for Symbolic Logic 2016 

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