Implications between Induction Principles for $\mathbb{N}$ in Peano Arithmetic

TL;DR

This paper critically examines the logical relationships between various induction principles in Peano arithmetic, revealing that some implications only hold under additional assumptions, thus challenging common equivalence claims.

Contribution

It clarifies which induction principles are truly equivalent in Peano arithmetic and highlights the necessity of extra conditions for certain implications.

Findings

Some implications between induction principles require the assumption that every nonzero natural number is a successor.

Counterexamples show invalid implications when structures do not satisfy the full Peano axioms.

The paper identifies valid and almost valid implications among nine induction-related properties.

Abstract

In introductory books about natural numbers, a common kind of assertion - often left as exercise to the reader - is that certain forms of induction on $\mathbb{N}$ (regular/ordinary, complete/strong) are equivalent one to each other and to the well-ordering principle. This means that if P1 and P2 are two of these principles, then, under all the other usually adopted postulates for the set of natural numbers (e.g., Peano axioms other than the induction axiom), P1 implies P2 and vice-versa. In this paper, we shows that, for a reasonable formalization, based on Peano arithmetic, some of the alleged implications between these principles hold only if an additional, independent condition is assumed, namely: every nonzero natural number is a successor. This condition is a consequence of the regular induction principle, but not of other induction principles. So, it is necessary to review all currently accepted implications between induction principles and similar statements in $\mathbb{N}$. From a list of 9 properties, usually considered induction principles (or equivalent to), we identify all valid and "almost valid" implications between them, as well as disprove all invalid implications by counterexamples, which consist of structures similar to Peano models, but not necessarily satisfying the Peano's induction axiom, and having an order relation adequately subordinated to the zero and the successors. Final remarks are made about what can and what cannot be proved if we weaken the assumptions about the order relation. The aim of this paper is to shed light on the need for more caution when alleging equivalence between induction principles for $\mathbb{N}$.