The set-membership symbol in every maths textbook began as Peano's Greek epsilon
Counting feels too obvious to need rules, yet in 1889 Giuseppe Peano set down a short list of axioms from which the arithmetic of whole numbers could be rebuilt using just zero and a 'next number' function. His notation mostly flopped, but one squiggle survived: the sign mathematicians still use for belonging to a set.
Peano was not first. In the 1860s Hermann Grassmann showed that many arithmetic facts follow from basic truths about succession and induction; Charles Sanders Peirce offered an axiom system in 1881 and Richard Dedekind another in 1888. Peano's contribution, published in Latin as Arithmetices principia, nova methodo exposita, was a streamlined version, which is why the scheme is also called the Dedekind-Peano axioms. Logic notation was young then. Peano kept mathematical and logical symbols strictly apart, a separation Gottlob Frege had introduced in his 1879 Begriffsschrift, but Peano did not know Frege's work and rebuilt the machinery from Boole and Schroder instead.
The list has nine axioms. One says a first natural number exists; Peano originally began at 1, while his later Formulario mathematico included zero. Four more concern equality and are now usually treated as general logic rather than arithmetic. Three pin down the successor function S, so that 1 is S(0), 2 is S(S(0)) and so on. Those alone do not guarantee that repeating S from zero reaches every number, and that gap is closed by the ninth axiom, induction.
Peano stated induction as a second-order principle, talking about all properties at once. Modern practice often swaps it for a weaker first-order scheme and adds addition and multiplication as extra axioms, and the name Peano arithmetic sometimes refers specifically to that restricted system. With the full second-order version, addition, multiplication and ordering can be defined directly: a plus S(b) is simply S(a plus b), and a is at most b when some c added to a gives b.
From these few rules the familiar structure emerges. Addition is commutative with zero as its identity, multiplication distributes over addition, and since nothing sits between 0 and 1 the naturals form a discrete ordered semiring. A strengthened form of induction even proves that every nonempty set of naturals has a smallest member. The axioms have barely changed since, and they became the testing ground for questions about whether number theory is consistent and complete.
Source: Peano axioms
