Is induction a circular way to define natural numbers?

[Evgeny.Makarov]

Sorry about the intriguing title; this is just a continuation of the discussion in https://driven2services.com/staging/mh/index.php?threads/5216/ from the Discrete Math forum. The original question there was how to introduce mathematical induction in a clear and convincing way. Since the current discussion about the foundations of mathematics is clearly off-topic, I decided to continue it in a separate thread.

I'm sure you are aware how the set of finite ordinals is constructed. So why is there a contradiction?

Obviously, there is no formal contradiction in Peano arithmetic or in the set-theoretic construction of natural numbers, or at least none has been found yet. The question is about a philosophical justification of Peano arithmetic.

Here is a quotation from Daniel Leivant, Intrinsic Logic and Computational Complexity, in LNCS 960, p. 192.

"The set $\mathbb{N}$ of natural numbers is implicitly defined by Peano's axioms: the generative axioms [$0\in\mathbb{N}$ and $n\in\mathbb{N}\to Sn\in\mathbb{N}$] convey a lower bound on the extension of $\mathbb{N}$, and the induction schema approximates the upper bound. However, as observed in (Edward Nelson, Predicative Arithmetic, Princeton University Press, 1986), if a formula $\varphi$ has quantifiers, then its meaning presupposes the delineation of $\mathbb{N}$ as the domain of the quantifiers, and therefore using induction over $\varphi$ as a component of the delineation of $\mathbb{N}$ is a circular enterprise."

As I said, I don't claim that I fully understand this.

We can prove the Peano axioms in this set, from set theory. That means that there are natural numbers (let's not focus on what "are" means :D ). Now, is the problem proving the uniqueness of a Peano model, modulo isomorphism?

Peano axioms (a first-order theory) has infinitely many non-isomorphic models (a corollary of the compactness theorem). However, it is easy to construct a single second-order formula whose only model are natural numbers.

 

[ModusPonens]

Hello

Sorry for my poor choice of words in a discussion about mathematics. I meant "where is the circularity?", not contradiction.

I may be way out of my league, but my question is the following: there are the Peano axioms. They don't define the natural numbers. It seems to me that you have a model which fits the axioms.