The Status of Mathematical Induction in an Axiomatic System

Abstract

This thesis investigates the status of Mathematical Induction (MI) in an axiomatic system. It first reviews and analyses the status of MI in the works of Gotlob Frege and Richard Dedekind, the pioneers of logicism who, in providing foundations for arithmetic, attempted to reduce MI to what they considered logic to be. These analyses reveal that their accounts of MI have the same structure and produce the same result. This is true even though the two thinkers used different components as fundamental logical elements and went through different routes to eventually prove (on the basis of more fundamental logical axioms and rules of inference and definitions) what they considered MI to be. Based on these analyses, we infer a formulation, i.e., U-MI, that presents both Frege’s and Dedekind’s formulations of MI. We then evaluate the possible proof- and model-theoretic problems that such a formulation of MI faces. These problems, among others, include certain difficulties with U-MI as a representation of mathematical induction, the problem of impredicativity, and the unattainability of the infinitary nature of MI in a finitary logic. We then introduce and defend our own account of the status of MI in an axiomatic system, in which MI is axiomatizable/derivable in an infinitary many-sorted logic. The final part of the study investigates concerns with the metatheoretical use of MI – in particular the circularity problem in such a use. Within this last part, we also explicate and elaborate on one of the advantages of our account of the status of MI in an axiomatic system in comparison to the rival accounts.