In mathematical logic, the Peano axioms (/piˈɑːnoʊ/,^[1] [peˈaːno]), also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th-century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete.

The axiomatization of arithmetic provided by Peano axioms is commonly called Peano arithmetic.

The importance of formalizing arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction.^[2][3] In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic.^[4][5] In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms in his book The principles of arithmetic presented by a new method (Latin: Arithmetices principia, nova methodo exposita).

The nine Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set of natural numbers. The next four are general statements about equality; in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the "underlying logic".^[6] The next three axioms are first-order statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final axiom is a second-order statement of the principle of mathematical induction over the natural numbers, which makes this formulation close to second-order arithmetic. A weaker first-order system is obtained by explicitly adding the addition and multiplication operation symbols and replacing the second-order induction axiom with a first-order axiom schema. The term Peano arithmetic is sometimes used for specifically naming this restricted system.

Historical second-order formulationedit

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When Peano formulated his axioms, the language of mathematical logic was in its infancy. The system of logical notation he created to present the axioms did not prove to be popular, although it was the genesis of the modern notation for set membership (∈, which comes from Peano's ε). Peano maintained a clear distinction between mathematical and logical symbols, which was not yet common in mathematics; such a separation had first been introduced in the Begriffsschrift by Gottlob Frege, published in 1879.^[7] Peano was unaware of Frege's work and independently recreated his logical apparatus based on the work of Boole and Schröder.^[8]

The Peano axioms define the arithmetical properties of natural numbers, usually represented as a set N or ${\displaystyle \mathbb {N} .}$ The non-logical symbols for the axioms consist of a constant symbol 0 and a unary function symbol S.

The first axiom states that the constant 0 is a natural number:

Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number,^[9] while the axioms in Formulario mathematico include zero.^[10]

The next four axioms describe the equality relation. Since they are logically valid in first-order logic with equality, they are not considered to be part of "the Peano axioms" in modern treatments.^[8]

The remaining axioms define the arithmetical properties of the natural numbers. The naturals are assumed to be closed under a single-valued "successor" function S.

Axioms 1, 6, 7, 8 define a unary representation of the intuitive notion of natural numbers: the number 1 can be defined as S(0), 2 as S(S(0)), etc. However, considering the notion of natural numbers as being defined by these axioms, axioms 1, 6, 7, 8 do not imply that the successor function generates all the natural numbers different from 0.

The intuitive notion that each natural number can be obtained by applying successor sufficiently many times to zero requires an additional axiom, which is sometimes called the axiom of induction.

The induction axiom is sometimes stated in the following form:

In Peano's original formulation, the induction axiom is a second-order axiom. It is now common to replace this second-order principle with a weaker first-order induction scheme. There are important differences between the second-order and first-order formulations, as discussed in the section § Peano arithmetic as first-order theory below.

Defining arithmetic operations and relationsedit

If we use the second-order induction axiom, it is possible to define addition, multiplication, and total (linear) ordering on N directly using the axioms. However, with first-order induction, this is not possible^{[citation needed]} and addition and multiplication are often added as axioms. The respective functions and relations are constructed in set theory or second-order logic, and can be shown to be unique using the Peano axioms.

Additionedit

Addition is a function that maps two natural numbers (two elements of N) to another one. It is defined recursively as:

${\displaystyle {\begin{aligned}a+0&=a,&{\textrm {(1)}}\\a+S(b)&=S(a+b).&{\textrm {(2)}}\end{aligned}}}$

For example:

${\displaystyle {\begin{aligned}a+1&=a+S(0)&{\mbox{by definition}}\\&=S(a+0)&{\mbox{using (2)}}\\&=S(a),&{\mbox{using (1)}}\\\\a+2&=a+S(1)&{\mbox{by definition}}\\&=S(a+1)&{\mbox{using (2)}}\\&=S(S(a))&{\mbox{using }}a+1=S(a)\\\\a+3&=a+S(2)&{\mbox{by definition}}\\&=S(a+2)&{\mbox{using (2)}}\\&=S(S(S(a)))&{\mbox{using }}a+2=S(S(a))\\{\text{etc.}}&\\\end{aligned}}}$

To prove commutativity of addition, first prove ${\displaystyle 0+b=b}$ and ${\displaystyle S(a)+b=S(a+b)}$, each by induction on ${\displaystyle b}$. Using both results, then prove ${\displaystyle a+b=b+a}$ by induction on ${\displaystyle b}$. The structure (N, +) is a commutative monoid with identity element 0. (N, +) is also a cancellative magma, and thus embeddable in a group. The smallest group embedding N is the integers.^{[citation needed]}

Multiplicationedit

Similarly, multiplication is a function mapping two natural numbers to another one. Given addition, it is defined recursively as:

${\displaystyle {\begin{aligned}a\cdot 0&=0,\\a\cdot S(b)&=a+(a\cdot b).\end{aligned}}}$

It is easy to see that ${\displaystyle S(0)}$ is the multiplicative right identity:

${\displaystyle a\cdot S(0)=a+(a\cdot 0)=a+0=a}$

To show that ${\displaystyle S(0)}$ is also the multiplicative left identity requires the induction axiom due to the way multiplication is defined:

• ${\displaystyle S(0)}$ is the left identity of 0: ${\displaystyle S(0)\cdot 0=0}$.
• If ${\displaystyle S(0)}$ is the left identity of ${\displaystyle a}$ (that is ${\displaystyle S(0)\cdot a=a}$), then $$Zdroj:https://en.wikipedia.org?pojem=Peano_arithmetic
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