Natural numbers and natural cardinals as abstract objects: A partial reconstruction of Frege's Grundgesetze in object theory

Journal of Philosophical Logic 28 (6):619-660 (1999)
Abstract
In this paper, the author derives the Dedekind-Peano axioms for number theory from a consistent and general metaphysical theory of abstract objects. The derivation makes no appeal to primitive mathematical notions, implicit definitions, or a principle of infinity. The theorems proved constitute an important subset of the numbered propositions found in Frege's Grundgesetze. The proofs of the theorems reconstruct Frege's derivations, with the exception of the claim that every number has a successor, which is derived from a modal axiom that logicians implicitly accept. In the final section of the paper, there is a brief philosophical discussion of how the present theory relates to the work of other philosophers attempting to reconstruct Frege's conception of numbers and logical objects
Keywords abstract objects  cardinal numbers  Dedekind  Frege  Hume  natural numbers  Peano
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DOI 10.1023/A:1004330128910
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References found in this work BETA
Logic, Logic, and Logic.George Boolos - 1998 - Harvard University Press.
The Foundations of Arithmetic.Gottlob Frege - 1953 - Evanston: Ill., Northwestern University Press.
Abstract Objects.Bob Hale - 1988 - Blackwell.
The Limits of Abstraction.Kit Fine - 2002 - Oxford University Press.

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Citations of this work BETA
Steps Toward a Computational Metaphysics.Branden Fitelson & Edward N. Zalta - 2007 - Journal of Philosophical Logic 36 (2):227-247.
Bennett and “Proxy Actualism”.Michael Nelson & Edward N. Zalta - 2009 - Philosophical Studies 142 (2):277-292.
Frege, Boolos, and Logical Objects.David J. Anderson & Edward N. Zalta - 2004 - Journal of Philosophical Logic 33 (1):1-26.

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