Generalizing boolos’ theorem

Review of Symbolic Logic 10 (1):80-91 (2017)
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Abstract

It’s well known that it’s possible to extract, from Frege’s Grudgesetze, an interpretation of second-order Peano Arithmetic in the theory  HP2, whose sole axiom is Hume’s principle. What’s less well known is that, in Die Grundlagen Der Arithmetic §82–83 Boolos (2011), George Boolos provided a converse interpretation of HP2 in PA2 . Boolos’ interpretation can be used to show that the Frege’s construction allows for any model of PA2 to be recovered from some model of HP2. So the space of possible arithmetical universes is precisely characterized by Hume’s principle. In this paper, I show that a large class of second-order theories admit characterization by an abstraction principle in this sense. The proof makes use of structural abstraction principles, a class of abstraction principles of considerable intrinsic interest, and categories of interpretations in the sense of Visser (2003).

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Graham Leach-Krouse
Kansas State University

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References found in this work

Frege’s Theorem: An Introduction.Richard G. Heck - 1999 - The Harvard Review of Philosophy 7 (1):56-73.
Finite axiomatizability using additional predicates.W. Craig & R. L. Vaught - 1958 - Journal of Symbolic Logic 23 (3):289-308.
Second order logic or set theory?Jouko Väänänen - 2012 - Bulletin of Symbolic Logic 18 (1):91-121.
Logic, Logic, and Logic.Vann McGee - 2001 - Bulletin of Symbolic Logic 7 (1):58-62.

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