In the book "Joseph J. Rotman Advanced Modern Algebra" the induction principle is derived by the principle of minimum but not using an axiomatic system of natural numbers. Is it possible to have axioms for natural numbers where we don't have induction principle (used instead in Peano's axioms)?
I report also this quote that can be useful for this question:
It is mistakenly printed in several books and sources that the well-ordering principle is equivalent to the induction axiom. In the context of the other Peano axioms, this is not the case, but in the context of other axioms, they can be equivalent.
The common mistake in many erroneous proofs is to assume that n-1 is a unique and well-defined natural number, a property which is not implied by the other Peano axioms.