A system of five axioms for the set of natural numbers and a function (successor) on it, introduced by G. Peano (1889):
5) for any property (axiom of induction).
In the first version 1 was used instead of 0. 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
can be combined to a single one:
if one defines as
The independence is proved by exhibiting a model on which all the axioms are true except one. For
such a model is the series of natural numbers beginning with 1; for
it is the set , where , ; for
the set ; for
the set with ; for
the set .
Sometimes one understands by Peano arithmetic the system in the first-order language with the function symbols , 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).
|||S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951)|
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.
|[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)|
Peano axioms. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Peano_axioms&oldid=11715