Peano axioms

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).

The axiom of induction (axiom 5) is a statement in second-order language. Dedekind proved that the system of Peano axioms with a second-order axiom of induction is categorical, that is, any two models $(\mathbf{N}, S, 0)$ and $(\mathbf{N}’, S', 0’)$ are mutually isomorphic. The isomorphism is determined by a function $f(x, y)$, where

$f(0,0) = 0’$, $f(Sx, Sx) = S’ f(x, x)$;
$f(x, Sy) = f(x, y)$; $f(x, y) = 0$ for $y < x$.

The existence of $f(x, y)$ for all pairs $(x, y)$ and the mutual single-valuedness for $x \leq y$ are proved by induction.

Peano's axioms make it possible to develop number theory and, 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 Peano’s axioms is proved by exhibiting, for each axiom, a model for which the axiom considered is false, but for which all the other axioms are true. For example:

• for axiom 1, such a model is the set 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 $\{0\}$
• for axiom 4, it is the set $\{0, 1\}$, with $S0 = S1 = 1$
• for axiom 5, it is the set $\mathbb{N} \cup \{-1\}$

Using this method, Peano provided a proof of independence for his axioms (1891).

Sometimes one understands by the term Peano arithmetic the system in the first-order language

with the function symbols
$S, +, \cdot$,
consisting of axioms
$Sx\neq 0$ and $Sx = Sy \to x = y$
defining equalities for $+$ and $\cdot$
$x + 0 = x$ and $x + Sx = S(x + y)$
$x \cdot 0 = 0$ and $x \cdot S(y) = x \cdot y + x$
and with the induction scheme
$A (0) \wedge \forall x (A(x) \to A(Sx)) \to \forall x A(x)$

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

References

• S.C. Kleene, Introduction to Metamathematics, North-Holland (1951).