Peano axioms

The Peano axioms (also Dedekind Peano axioms or the Peano postulates ) are a set of axioms which characterize the natural numbers and their properties. They were formulated in 1889 by Italian mathematician Giuseppe Peano and serve to this day as Standardformalisierung the arithmetic for meta-mathematical investigations. While the original version of Peano can be formalized in predicate logic the second stage, a weaker variant is used in first-order logic now usually referred to as Peano arithmetic. With the exception of representatives of Ultrafinitismus Peano arithmetic is widely recognized in mathematics as correct and consistent characterization of natural numbers. Other formalization of natural numbers which are related to the Peano arithmetic, the Robinson arithmetic and primitive recursive arithmetic.

Richard Dedekind proved in 1888 that all models of Peano arithmetic with induction axiom second stage are isomorphic to the standard model, that is, that the structure of the natural numbers is uniquely characterized up to designation. On the other hand, this does not apply to the first-stage formalization of the Löwenheim - Skolem theorem implies the existence of other models that satisfy the Peano axioms.

Axioms

Original formalization

Peano looked originally 1 as the smallest natural number. In his later version of the axioms that are listed in the following modern, he replaced by 0 1 The axioms then have the following form:

These axioms can verbalize as follows, where the operator n ' as "successor of n " is read:

The last axiom is called induction axiom, since the evidence based method of complete induction on it. It is equivalent to the statement that every set of natural numbers has a smallest element. Also, it guarantees that Peano's are recursive definitions of addition and multiplication on at all well defined:

The One Peano defined as a successor of zero:

From this definition follows the definition of addition for the successor.

Peano continued as a framework requires a class logic. His axiom system is also in the set theory to interpret or even in the predicate second-order logic, since in addition to number of variables in the induction axiom, the quantity variable occurs.

Formalization in first-order logic

The original formalization in the induction axiom contains a quantification over sets of objects. As in the first-order logic can not be quantified over sets of objects, the induction axiom is replaced by a weaker axiom schema in the first order predicate logic for the formalization in the logic of the first stage. This has the following form:

• For all formulas

For each formula, the corresponding induction axiom must be added; the first stage version of Peano arithmetic thus contains an infinite set of axioms.