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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 eachother and to the well-ordering principle. This means that if P1 and P2 are twoof these principles, then, under all the other usually adopted postulates forthe 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 reasonableformalization, based on Peano arithmetic, some of the alleged implicationsbetween these principles hold only if an additional, independent condition isassumed, namely: every nonzero natural number is a successor. This condition isa consequence of the regular induction principle, but not of other inductionprinciples. So, it is necessary to review all currently accepted implicationsbetween induction principles and similar statements in $\mathbb{N}$. From alist of 9 properties, usually considered induction principles (or equivalentto), we identify all valid and "almost valid" implications between them, aswell as disprove all invalid implications by counterexamples, which consist ofstructures similar to Peano models, but not necessarily satisfying the Peano'sinduction axiom, and having an order relation adequately subordinated to thezero and the successors. Final remarks are made about what can and what cannotbe proved if we weaken the assumptions about the order relation. The aim ofthis paper is to shed light on the need for more caution when allegingequivalence between induction principles for $\mathbb{N}$.

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