Natural Numbers and Natural Cardinals as Abstract Objects: A Partial Reconstruction of Frege's Grundgesetze in Object Theory
Author
Edward N. Zalta
Reference
Journal of Philosophical Logic, 28/6 (1999):
619-660
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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