Induction Models on \mathbb{N}

TL;DR

This paper generalizes induction models in mathematics and computer science, providing formal characterizations of their reductions and equivalences, thus advancing understanding of their logical relationships.

Contribution

It introduces a generalized framework for induction models, characterizes reductions among them, and formalizes their equivalence, extending prior limited models.

Findings

Existence and construction of S for given B and vice versa

Formal characterization of reduction among induction models

Capture of equivalence among different induction models

Abstract

Mathematical induction is a fundamental tool in computer science and mathematics. Henkin initiated the study of formalization of mathematical induction restricted to the setting when the base case B is set to singleton set containing 0 and a unary generating function S. The usage of mathematical induction often involves wider set of base cases and k-ary generating functions with different structural restrictions. While subsequent studies have shown several Induction Models to be equivalent, there does not exist precise logical characterization of reduction and equivalence among different Induction Models. In this paper, we generalize the definition of Induction Model and demonstrate existence and construction of S for given B and vice versa. We then provide a formal characterization of the reduction among different Induction Models that can allow proofs in one Induction Models to be expressed as proofs in another Induction Models. The notion of reduction allows us to capture equivalence among Induction Models.