Peano's nine axioms of arithmetic

Statement

In 1889, Giuseppe Peano presented his The principles of arithmetic, presented by a new method, based on the work of Dedekind. Soare proposes that the origination of "primitive recursion" began formally with the axioms of Peano, although

"Well before the nineteenth century mathematicians used the principle of defining a function by induction. Dedekind 1888 proved, using accepted axioms, that such a definition defines a unique function, and he applied it to the definition of the functions m+n, m x n, and mⁿ. Based on this work of Dedekind, Peano 1889 and 1891 wrote the familiar five [sic] axioms for the positive integers. As a companion to his fifth [sic] axiom, mathematical induction, Peano used definition by induction, which has been called primitive recursion (since Péter 1934 and Kleene 1936) ... ."

Observe that in fact Peano's axioms are 9 in number and axiom 9 is the recursion/induction axiom.

"Subsequently the 9 were reduced to 5 as "Axioms 2, 3, 4 and 5 which deal with identity, belong to the underlying logic. This leaves the five axioms that have become universally known as "the Peano axioms ... Peano acknowledges (1891b, p. 93) that his axioms come from Dedekind ... ."