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Comparing Peano arithmetic, Basic Law V, and Hume’s Principle

Walsh, Sean (2012) Comparing Peano arithmetic, Basic Law V, and Hume’s Principle. Annals of Pure and Applied Logic 163 (11), pp. 1679-1709. ISSN 0168-0072.

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Official URL: http://dx.doi.org/10.1016/j.apal.2011.12.016

Abstract

This paper presents new constructions of models of Hume’s Principle and Basic Law V with restricted amounts of comprehension. The techniques used in these constructions are drawn from hyperarithmetic theory and the model theory of fields, and formalizing these techniques within various subsystems of second-order Peano arithmetic allows one to put upper and lower bounds on the interpretability strength of these theories and hence to compare these theories to the canonical subsystems of second-order arithmetic. The main results of this paper are: (i) there is a consistent extension of the hypearithmetic fragment of Basic Law V which interprets the hyperarithmetic fragment of second-order Peano arithmetic, and (ii) the hyperarithmetic fragment of Hume’s Principle does not interpret the hyperarithmetic fragment of second-order Peano arithmetic, so that in this specific sense there is no predicative version of Frege’s Theorem.

Item Type: Article
Keyword(s) / Subject(s): Second-order arithmetic, Basic Law V, Hume’s Principle, Hyperarithmetic, Recursively saturated, Interpretability
School or Research Centre: Birkbeck Schools and Research Centres > School of Social Sciences, History and Philosophy > Philosophy
Depositing User: Administrator
Date Deposited: 16 Jan 2012 12:06
Last Modified: 13 Jun 2013 09:45
URI: http://eprints.bbk.ac.uk/id/eprint/4556

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