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There is another version of the Peano axioms, usually known as the second-order Peano axioms, with the property that there is only one mathematical structure satisfying them (namely \(\mathbb {N}\)), which can be used as a definition of \(\mathbb {N}\). In contrast, there are many nonisomorphic structures, known as nonstandard models, that satisfy the first-order Peano axioms.
As far as the consistency of first-order arithmetic is concerned, the distinction between intuitionistic logic and classical logic turns out not to matter too much. Gödel, and independently Gentzen [13], showed constructively that Heyting arithmetic, which is the intuitionistic counterpart of PA, is consistent if and only PA is consistent.
Note in particular that the variables are not allowed to range over sets of elements; this restriction is what makes PA a first-order theory.
For much more historical context, I recommend the article by Kahle [10].
On the other hand, some people, such as Solomon Feferman [5], explicitly reject platonism but nevertheless find the argument that \(\mathbb {N}\) satisfies all the axioms of PA to be completely convincing.
See, for example, Willard [18] for an explanation of how this might be possible.
In fact, the theorem can be further weakened to assert the stabilization of all primitive recursive descending sequences of ordinals; see [17, Lemma 12.79] or [2, Theorem 4.6], for example. The fact that PRA plus Theorem 2 implies that PA is consistent is only implicit and not explicit in Gentzen’s original proof. I have chosen this way of presenting the argument rather than the more common approach of explaining what “induction up to \(\epsilon _0\)” is, because I believe that Theorem 2 is more accessible to the general reader without training in logic and set theory.
A far stronger induction argument was used by Robertson and Seymour in their proof of the graph minor theorem [9], and nobody seems to have rejected that theorem on those grounds.
Note that the unreasonable soundness of mathematics is not the same as Eugene Wigner’s unreasonable effectiveness of mathematics in the natural sciences. What Stefan cannot explain are mathematicians’ purely mathematical predictions rather than their scientific predictions.
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Chow, T.Y. The Consistency of Arithmetic. Math Intelligencer 41, 22–30 (2019). https://doi.org/10.1007/s00283-018-9837-z
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DOI: https://doi.org/10.1007/s00283-018-9837-z