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- Jan 30, 2012
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Sorry about the intriguing title; this is just a continuation of the discussion in this thread 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.
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.
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.
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.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). Now, is the problem proving the uniqueness of a Peano model, modulo isomorphism?
