Peano Axioms - The Axioms

The Axioms

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 is from Peano's ε) and implication (⊃, which is 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 signature (a formal language's non-logical symbols) for the axioms includes a constant symbol 0 and a unary function symbol S.

The constant 0 is assumed to be a natural number:

1. 0 is a natural number.

The next four axioms describe the equality relation.

2. For every natural number x, x = x. That is, equality is reflexive.
3. For all natural numbers x and y, if x = y, then y = x. That is, equality is symmetric.
4. For all natural numbers x, y and z, if x = y and y = z, then x = z. That is, equality is transitive.
5. For all a and b, if a is a natural number and a = b, then b 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.

6. For every natural number n, S(n) is a natural number.

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 and 6 define a unary representation of the natural numbers: the number 1 can be defined as S(0), 2 as S(S(0)) (which is also S(1)), and, in general, any natural number n as Sn(0). The next two axioms define the properties of this representation.

7. For every natural number n, S(n) = 0 is false. That is, there is no natural number whose successor is 0.
8. For all natural numbers m and n, if S(m) = S(n), then m = n. That is, S is an injection.

Axioms 1, 6, 7 and 8 imply that the set of natural numbers contains the distinct elements 0, S(0), S(S(0)), and furthermore that {0, S(0), S(S(0)), …} ⊆ N. This shows that the set of natural numbers is infinite. However, to show that N = {0, S(0), S(S(0)), …}, it must be shown that N ⊆ {0, S(0), S(S(0)), …}; i.e., it must be shown that every natural number is included in {0, S(0), S(S(0)), …}. To do this however requires an additional axiom, which is sometimes called the axiom of induction. This axiom provides a method for reasoning about the set of all natural numbers.

9. If K is a set such that:
   • 0 is in K, and
   • for every natural number n, if n is in K, then S(n) is in K,
   then K contains every natural number.

The induction axiom is sometimes stated in the following form:

9. If φ is a unary predicate such that:
   • φ(0) is true, and
   • for every natural number n, if φ(n) is true, then φ(S(n)) is true,
   then φ(n) is true for every natural number n.

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.