Peano axioms

The Peano axioms (also Dedekind-Peano axioms or Peano postulates ) are five axioms that characterize the natural numbers and their properties. They were formulated in 1889 by the Italian mathematician Giuseppe Peano and still serve as the standard formalization of arithmetic for metamathematic investigations. While the original version of Peano can be formalized in second-order predicate logic , a weaker variant in first-order predicate logic known as Peano arithmetic is mostly used today . With the exception of representatives of ultrafinitism , Peano arithmetic is generally recognized in mathematics as the correct and consistent characterization of natural numbers. Other formalizations of the natural numbers that are related to Peano arithmetic are Robinson arithmetic and primitive recursive arithmetic .

Richard Dedekind proved as early as 1888 Dedekind's so-called isomorphism theorem , that all models of Peano arithmetic with the induction axiom of the second order are isomorphic to the standard model , i.e. This means that the structure of the natural numbers is uniquely characterized except for naming. However, this does not apply to the first-level formalization, from the Löwenheim-Skolem theorem follows the existence of pairwise non-isomorphic models (including models of every infinite cardinality) that satisfy the Peano axioms. ${\ displaystyle \ mathbb {N}}$

Axioms

Original formalization

Peano originally considered 1 to be the smallest natural number. In his later version of the axioms, which are notated in a modern way below, he replaced 1 with 0. The axioms then have the following form:

1. ${\ displaystyle 0 \ in \ mathbb {N}}$
2. ${\ displaystyle \ forall n (n \ in \ mathbb {N} \ Rightarrow n '\ in \ mathbb {N})}$
3. ${\ displaystyle \ forall n (n \ in \ mathbb {N} \ Rightarrow n '\ not = 0)}$
4. ${\ displaystyle \ forall n, m (m, n \ in \ mathbb {N} \ Rightarrow (m '= n' \ Rightarrow m = n))}$
5. ${\ displaystyle \ forall X (0 \ in X \ land \ forall n (n \ in \ mathbb {N} \ Rightarrow (n \ in X \ Rightarrow n '\ in X)) \ Rightarrow \ mathbb {N} \ subseteq X)}$

These axioms can be verbalized as follows, reading “ successors of ”: ${\ displaystyle n '}$${\ displaystyle n}$

1. 0 is a natural number.
2. Every natural number has a natural number as a successor.${\ displaystyle n}$${\ displaystyle n '}$
3. 0 is not a natural number successor.
4. Natural numbers with the same successor are the same.
5. If the 0 contains and, with every natural number, also its successor , the natural numbers form a subset of .${\ displaystyle X}$${\ displaystyle n}$${\ displaystyle n '}$${\ displaystyle X}$

The last axiom is called the axiom of induction because it is the basis of the complete induction proof method . It is equivalent to saying that every nonempty set of natural numbers has a smallest element. It also guarantees that Peano's recursive definitions of addition and multiplication are well-defined at all : ${\ displaystyle \ mathbb {N}}$

${\ displaystyle n + 0: = n \,}$
${\ displaystyle n + m ': = (n + m)' \,}$
${\ displaystyle n \ cdot 0: = 0}$
${\ displaystyle n \ cdot m ': = (n \ cdot m) + n}$

Peano defined one as the successor to zero:

${\ displaystyle 1: = 0 '\,}$

From this definition follows with the addition definition for the successor . ${\ displaystyle \, n '= n + 1}$

Peano assumed a class logic as a framework . His system of axioms can also be interpreted in set theory or in second-order predicate logic , since the induction axiom also includes number variables as well as set variables . ${\ displaystyle X}$

Formalization in first-order predicate logic

In the axiom of induction, the original formalization contains a quantification over sets of objects (see above). However, since the first-level predicate logic cannot quantify using sets of objects, the induction axiom is replaced by a weaker axiom scheme in the first-level predicate logic for the formalization in the first-level logic. This has the following form:

• ${\ displaystyle (\ phi (0) \ land \ forall n (\ phi (n) \ Rightarrow \ phi (n '))) \ Rightarrow \ forall n \ phi (n)}$ for all formulas ${\ displaystyle \ phi (x)}$
If the validity of holds for every number n , then the formula holds for every natural number n.${\ displaystyle \ phi (0)}$${\ displaystyle \ phi (n)}$${\ displaystyle \ phi (n ')}$${\ displaystyle \ phi (n)}$

For each formula the corresponding induction axiom must be added; the first-level version of Peano arithmetic thus contains an infinite set of axioms. ${\ displaystyle \ phi (x)}$

Individual evidence

1. a b Peano: Arithmetices principia nova methodo exposita, Turin 1889
2. Peano: Opere scelte III, p. 216, original with operator n + instead of n '
3. ^ Peano: Opere scelte III, pp. 221 and 229
4. ^ Peano: Opere scelte III, p. 220