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A system of five axioms for the set of natural numbers $\mathbb{N}$ and a function $S$ (successor) on it, introduced by G. Peano (1889):

1. $0 \in \mathbb{N}$
2. $x \in \mathbb{N} \to Sx \in \mathbb{n}$
3. $x \in \mathbb{N} \to Sx \neq 0$
4. $x \in \mathbb{N} \wedge y \in \mathbb{N} \wedge Sx =Sy \to x = y$
5. $0 \in M \wedge \forall x (x\in M \to Sx\in M) \to \mathbb{N} \subseteq M$ for any property $M$ (axiom of induction).

In the first version of his system, Peano used $1$ instead of $0$ in Axioms 1, 3, and 5. Similar axioms were proposed by R. Dedekind (1888).

Peano's axioms are categorical, that is, any two systems and satisfying them are isomorphic. The isomorphism is determined by a function , where

The existence of for all pairs and the mutual single-valuedness for are proved by induction.

Peano's axioms make it possible to develop number theory, in particular, to introduce the usual arithmetic functions and to establish their properties.

All the axioms are independent, but

and

can be combined to a single one:

if one defines as

The independence of the axioms is proved by exhibiting, for each axiom, a model for which the other axioms are true, but the one being considered is false:

• For Axiom 1, such a model is the series of natural numbers beginning with $1$
• For Axiom 2, it is the set $\mathbb{N} \cup \{1/2\}$, with $S0 = 1/2 and S1/2 =1$
• For Axiom 3, it is the set $\{0\}$, with $S0 = 0$
• For Axiom 4, it is the set $\{0, 1\}$, with $S0 = S1 = 1$
• For Axiom 5, it is the set $\mathbb{N} \cup \{-1\}$, with $S-1 = -1$

Sometimes one understands by Peano arithmetic the system in the first-order language with the function symbols $S, +, *$, consisting of the axioms

defining equalities for and , and the induction scheme

where is an arbitrary formula, known as the induction formula (see Arithmetic, formal).

References

[1]	S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951)

Comments

The system of Peano arithmetic mentioned at the end of the article above is no longer categorical (cf. also Categoric system of axioms), and gives rise to so-called non-standard models of arithmetic.

References

[a1]	H.C. Kennedy, "Peano. Life and works of Giuseppe Peano" , Reidel (1980)
[a2]	H.C. Kennedy, "Selected works of Giuseppe Peano" , Allen & Unwin (1973)
[a3]	E. Landau, "Grundlagen der Analysis" , Akad. Verlagsgesellschaft (1930)