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

In this paper, the author derives the Dedekind-Peano axioms for number theo ry from a consistent and general metaphysical theory of abstract objects. T he derivation makes no appeal to primitive mathematical notions, implicit d efinitions, or a principle of infinity. The theorems proved constitute an i mportant subset of the numbered propositions found in Frege's Grundgesetze. The proofs of the theorems reconstruct Frege's derivations, with the excep tion of the claim that every number has a successor, which is derived from a modal axioms that (philosophical) logicians implicitly accept. In the fin al section of the paper, there is a brief philosophical discussion of how t he present theory relates to the work of other philosophers attempting to r econstruct Frege's conception of numbers and logical objects.