Peano, Giuseppe See LOGICAL FORM, PEANO POS -. TULATE. Peano postulates, also called Peano axioms, a list of assumptions from which the integers can be defined from some initial integer, equality, and successorship, and usually seen as defining progressions. The Peano postulates for arithmetic were produced by G. Peano in 1889. He took the set N of integers with a first term 1 and an equality relation between them, and assumed these nine axioms: 1 belongs to N; N has more than one member; equality is reflexive, symmetric, and associative, and closed over N; the successor of any integer in N also belongs to N, and is unique; and a principle of mathematical induction applying across the members of N, in that if 1 belongs to some subset M of N and so does the successor of any of its members, then in fact M % N. In some ways Peano’s formulation was not clear. He had no explicit rules of inference, nor any guarantee of the legitimacy of inductive definitions (which Dedekind established shortly before him). Further, the four properties attached to equality were seen to belong to the underlying ‘logic’ rather than to arithmetic itself; they are now detached.
It was realized (by Peano himself) that the postulates specified progressions rather than integers (e.g., 1, ½, ¼, 1/8, . . . , would satisfy them, with suitable interpretations of the properties). But his work was significant in the axiomatization of arithmetic; still deeper foundations would lead with Russell and others to a major role for general set theory in the foundations of mathematics.
In addition, with O. Veblen, T. Skolem, and others, this insight led in the early twentieth century to ‘non-standard’ models of the postulates being developed in set theory and mathematical analysis; one could go beyond the ‘. . .’ in the sequence above and admit ‘further’ objects, to produce valuable alternative models of the postulates. These procedures were of great significance also to model theory, in highlighting the property of the non-categoricity of an axiom system. A notable case was the ‘non-standard analysis’ of A. Robinson, where infinitesimals were defined as arithmetical inverses of transfinite numbers without incurring the usual perils of rigor associated with them. See also PHILOSOPHY OF MATHEMATICS. I.G.-G.