3
$\begingroup$

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

$\endgroup$
9
  • 1
    $\begingroup$ Have you read cited reference 21, where your question is thoroughly explored? $\endgroup$
    – Eric Towers
    Dec 5, 2019 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, 2019 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, 2019 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, 2019 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, 2019 at 16:11

1 Answer 1

Reset to default
0
$\begingroup$

Some axiomatizations appearing in documents, roughly arranged in increasing order of reliance on induction:

  • As mentioned in the comments, there is an "inductive set" definition of the naturals, from which induction follows. From this, it follows that every natural number is either 0 or a successor. Welch states this as proposition 2.6 after proving the correctness of induction along $\omega$ as theorem 2.5, by showing that the set $\{n\in\omega\mid\Phi(n)\}$ must be inductive, and an inductive subset of $\omega$ is equal to $\omega$.
  • T. L. Wong has a set of lecture notes titled "Model theory of arithmetic", in which a theory $\mathrm{PA}^-$ is introduced, including axiom (xiii): $\forall x,y(x<y\implies\exists z(y=x+z+1))$. Here, we are able to guarantee that for nonzero $y$, $y-1$ exists and is unique.
  • In Slaman's paper $\Sigma_n$-bounding and $\Delta_n$-induction, an axiom similar to Wong's (xiii) is introduced, except the trailing $+1$ is removed. As $\omega$ is an additively principal ordinal, this rules out the case from "Are induction and well-ordering equivalent?" of $(\omega2,<)$.
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .