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The infinity of primary numbers: a case study of purity of methods. (L’infinité des nombres premiers: une étude de cas de la pureté des méthodes.) (French. English summary) Zbl 1406.03024
This is a very interesting study of the purity of proofs in the special case of the theorem on the infinity of primes, in the sense of their co-finality, that is, the statement that for any \(n\) there exists a prime number \(p\) with \(p>n\). The author’s aim is to show that while Euclid’s proof can be made to be pure (in the sense that it is based only on what is contained in the statement to be proved), the proof in [H. Furstenberg, Am. Math. Mon. 62, No. 5, 353 (1955; Zbl 1229.11009)] is not pure. After a survey of the fragments of Peano arithmetic in which Euclid’s proof of the infinity of primes can be formalized, the author addresses the possible critique regarding Euclid’s proof, that it uses addition, when the notion of a prime number can be understood in terms of multiplication only. One can either point out that natural numbers are usually considered endowed with a ring structure, or that one can define \(a+b=c\) in terms of \(S\) (the successor function) and \(\cdot\) in the manner of Julia Robinson as \(S (a\cdot c)\cdot S(b\cdot c) = S[(c\cdot c)\cdot S(a\cdot b)]\), or else – and that is certainly the most convincing answer – that the cofinality of primes can be proved inside an axiom system expressed in terms of \(S\), \(<\), \(\cdot\), consisting of 17 axioms, one of which is a strong form of the induction axiom. Axiom 7, stating that for every sequence of prime numbers \(p_1, \ldots, p_n\), there exists \(z\) such that \(z=p_1\cdot p_2\cdot \ldots\cdot p_n\), should have been stated more carefully. One needs more than an axiom schema stating that, for each numeral \(n\), the product \(z=p_1\cdot p_2\cdot \ldots\cdot p_n\) exists. One needs to know that the product of all primes less than any element \(a\) exists, so the axiom should have been stated as \((\forall x)(\exists z)(\forall p)\, \pi(p)\wedge p\leq a\rightarrow p\,|\, z\), where \(\pi(p)\) stands for \((\forall m)(\forall n)\, p=m\cdot n\rightarrow (m=1\vee n=1)\), and \(a\,|\, b\) stands for \((\exists c)\, a\cdot c=b\). Furstenberg’s proof, using topological notions, is considered impure, given that it uses notions that are not part of the statement of the co-finality of primes.

MSC:
03A05 Philosophical and critical aspects of logic and foundations
00A30 Philosophy of mathematics
03-03 History of mathematical logic and foundations
01A20 History of mathematics in Ancient Greece and Rome
01A60 History of mathematics in the 20th century
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