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Natural Numbers and Natural Cardinals as Abstract Objects: A Partial Reconstruction of Frege's "Grundgesetze" in Object Theory

Edward N. Zalta
Journal of Philosophical Logic
Vol. 28, No. 6 (Dec., 1999), pp. 619-660
Published by: Springer
Stable URL: http://www.jstor.org/stable/30226922
Page Count: 42
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Natural Numbers and Natural Cardinals as Abstract Objects: A Partial Reconstruction of Frege's
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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 (philosophical) 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.

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