When Peano formulated his axioms, the language of mathematical logic was in its infancy. The system of logical notation he created to present the axioms did not prove to be popular, although it was the genesis of the modern notation for set membership (∈, which comes from Peano's ε) and implication (⊃, which comes from Peano's reversed 'C'.) Peano maintained a clear distinction between mathematical and logical symbols, which was not yet common in mathematics; such a separation had first been introduced in the Begriffsschrift by Gottlob Frege, published in 1879. Peano was unaware of Frege's work and independently recreated his logical apparatus based on the work of Boole and Schröder.
The Peano axioms define the arithmetical properties of natural numbers, usually represented as a set N or The non-logical symbols for the axioms consist of a constant symbol 0 and a unary function symbol S.
The first axiom states that the constant 0 is a natural number:
The next four axioms describe the equality relation. Since they are logically valid in first-order logic with equality, they are not considered to be part of "the Peano axioms" in modern treatments.
- For every natural number x, x = x. That is, equality is reflexive.
- For all natural numbers x and y, if x = y, then y = x. That is, equality is symmetric.
- For all natural numbers x, y and z, if x = y and y = z, then x = z. That is, equality is transitive.
- For all a and b, if b is a natural number and a = b, then a is also a natural number. That is, the natural numbers are closed under equality.
The remaining axioms define the arithmetical properties of the natural numbers. The naturals are assumed to be closed under a single-valued "successor" function S.
- For every natural number n, S(n) is a natural number.
- For all natural numbers m and n, m = n if and only if S(m) = S(n). That is, S is an injection.
- For every natural number n, S(n) = 0 is false. That is, there is no natural number whose successor is 0.
Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number. This choice is arbitrary, as axiom 1 does not endow the constant 0 with any additional properties. However, because 0 is the additive identity in arithmetic, most modern formulations of the Peano axioms start from 0. Axioms 1, 6, 7, 8 define a unary representation of the intuitive notion of natural numbers: the number 1 can be defined as S(0), 2 as S(S(0)), etc. However, considering the notion of natural numbers as being defined by these axioms, axioms 1, 6, 7, 8 do not imply that the successor function generates all the natural numbers different from 0. Put differently, they do not guarantee that every natural number other than zero must succeed some other natural number.
The intuitive notion that each natural number can be obtained by applying successor sufficiently often to zero requires an additional axiom, which is sometimes called the axiom of induction.
- If K is a set such that:
then K contains every natural number.
- 0 is in K, and
- for every natural number n, n being in K implies that S(n) is in K,
The induction axiom is sometimes stated in the following form:
- If φ is a unary predicate such that:
then φ(n) is true for every natural number n.
- φ(0) is true, and
- for every natural number n, φ(n) being true implies that φ(S(n)) is true,
In Peano's original formulation, the induction axiom is a second-order axiom. It is now common to replace this second-order principle with a weaker first-order induction scheme. There are important differences between the second-order and first-order formulations, as discussed in the section § Models below.