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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[21] 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.[21]

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.[21]

Taken by: https://en.wikipedia.org/wiki/Mathematical_induction

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  • $\begingroup$ Have you read cited reference 21, where your question is thoroughly explored? $\endgroup$ – Eric Towers Dec 5 '19 at 16:01
  • $\begingroup$ It's noted here that the Peano axioms claim $\Bbb N$ is a discretely ordered semiring subject to the usual induction axiom schema. Since there are discretely ordered semirings other than $\Bbb N$, what additional properties would you want to characterise $\Bbb N$, beside induction? $\endgroup$ – J.G. Dec 5 '19 at 16:05
  • $\begingroup$ @EricTowers I saw, if I'm not wrong in the reference we see "common mistake in many erroneous proofs" but there's not an axiomatic system without induction. $\endgroup$ – asv Dec 5 '19 at 16:06
  • $\begingroup$ @J.G. In set theory the induction principle is derived from axioms, for example in page 19 here people.maths.bris.ac.uk/~mapdw/current-set-theory.pdf So maybe there's something we can formalize to have in axiomatic natural numbers a proof of induction. However I don't know very well set theory. $\endgroup$ – asv Dec 5 '19 at 16:10
  • $\begingroup$ @asv : Keep reading. Don't stop until you at least finish the section "Alternative Ways of Defining the Natural Numbers". $\endgroup$ – Eric Towers Dec 5 '19 at 16:11

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