Peano axioms

Giuseppe Peano.

The Peano axioms or Peano postulates are a system of second-order axioms for arithmetic devised by the mathematician Giuseppe Peano in the XIX, to define the natural numbers. These axioms have been used virtually unchanged in various mathematical investigations, including questions about the consistency and completeness of arithmetic and number theory.

He published them in 1889 in a booklet of about thirty pages, intitulated Aritmetices principia, nova method exposita, which is translated by New method of exposure of arithmetic principles. Gives a list of nine axioms, of which four are about the use of sign={displaystyle}". The others are known as "Axioms of Peano". The mathematicians consider them as the preliminary platform to forge the following usual sets of numbers. Peano's pivotal idea was that of "switch."

The Axioms

The five axioms or postulates of Peano are the following:

1. The 1{displaystyle 1} It's a natural number, then 1{displaystyle 1} is in the set N of natural numbers.
2. All natural numbers n{displaystyle n} has a successor n↓ ↓ {displaystyle n^{*}. (This axiom is used to define the sum later).
3. The 1{displaystyle 1} is not the successor of any natural number.
4. If there are two natural numbers n and m with the same successor, then n{displaystyle n} and m{displaystyle m} are the same natural number.
5. Yeah. 1{displaystyle 1} belongs to a set of natural numbers, and given any element, the successor also belongs to the set, then all natural numbers belong to that set. This last axiom is the principle of mathematical induction.

There is a debate on whether to consider 0{displaystyle} as a natural number or not. It is usually decided in each case, depending on whether it is needed or not. When it is resolved to include 0{displaystyle}, then some minor adjustments should be made:

1. The 0{displaystyle} It's a natural number.
2. Yeah. n{displaystyle n} is a natural number, then the successor of n{displaystyle n} It's also a natural number.
3. The 0{displaystyle} is not the successor of any natural number.
4. If there are two natural numbers n{displaystyle n} and m{displaystyle m} with the same successor, then n{displaystyle n} and m{displaystyle m} are the same natural number.
5. Yeah. 0{displaystyle} belongs to a set, and given any natural number, the successor of that number also belongs to that set, then all natural numbers belong to that set.

Formal presentation

As discussed above, there is a debate on whether to include 0{displaystyle} between natural numbers or not. Below are the axioms of Peano formally, contemplating both possibilities:

When zero is not involved

Symbols that designate primitive concepts are N,1,x♫{displaystyle ~N,1,x}.

The symbol N appoints a monadian preacher who reads “to be a natural number”. The symbol 1, for its part, designates a constant that aims to represent the number one. And the symbol x', finally, designates a function on x who returns the successor x. To this function many times you write it S(x). Finally, the metavariable φ φ {displaystyle phi } represents any formula of arithmetic, and φ φ (x){displaystyle phi (x)} represents any formula that has x as a free variable.

The five Peano axioms are:

A1:N(1){displaystyle A_{1}:N(1),}

A2:Русский Русский x(N(x)→ → N(x♫)){displaystyle A_{2}:forall x(N(x)to N(x))}}

A3:¬ ¬ consuming consuming x(N(x)∧ ∧ 1=x♫){displaystyle A_{3}:neg exists x(N(x)land 1=x')}}

A4:Русский Русский xРусский Русский and((N(x)∧ ∧ N(and)∧ ∧ x♫=and♫)→ → x=and){displaystyle A_{4}:forall xforall y(N(x)land N(y)land x'=y')to x=y)}

There are two variants of the fifth axiom. The first is formulated in first-order logic, and is actually an axiom schema. The second one is an axiom, but it is formulated in second order logic.

A5:(φ φ (1)∧ ∧ Русский Русский x(φ φ (x)→ → φ φ (x♫)))→ → Русский Русский xφ φ (x){displaystyle A_{5}:{Big (}phi (1)land forall x(phi (x)to phi (x)){Big)}to forall xphi (x)}

A5♫:Русский Русский φ φ ((φ φ (1)∧ ∧ Русский Русский x(φ φ (x)→ → φ φ (x♫)))→ → Русский Русский xφ φ (x)){displaystyle A_{5}':forall phi {bigg (}{Big (}{phi (1)land forall x(phi (x)to phi (x))}{Big)}to forall x phi (x){bigg)}}}}

In addition to the five axioms, Peano arithmetic uses two definitions (of addition and multiplication), which are sometimes presented as axioms. Here are all the variants:

Definitions of addition and multiplication:

D1:{displaystyle D_{1}:,}	n+1=n♫{displaystyle n+1=n',}
	n+m♫=(n+m)♫{displaystyle n+m'=(n+m)',}

D2:{displaystyle D_{2}:,}	n× × 1=n{displaystyle ntimes 1=n,}
	n× × m♫=(n× × m)+n{displaystyle ntimes m'=(ntimes m)+n,}

Axioms of addition and multiplication:

A6:{displaystyle A_{6}:,}	Русский Русский n(n+1=n♫){displaystyle forall n(n+1=n')}
	Русский Русский nРусский Русский m(n+m♫=(n+m)♫){displaystyle forall nforall m(n+m'=(n+m)'}

A7:{displaystyle A_{7}:,}	Русский Русский n(n× × 1=n){displaystyle forall n(ntimes 1=n),}
	Русский Русский nРусский Русский m(n× × m♫=(n× × m)+n){displaystyle forall nforall m(ntimes m'=(ntimes m)+n),}

When zero intervenes

Symbols that designate primitive concepts are N,0,x♫{displaystyle ~N,0,x}.

Axioms:

A1:N(0){displaystyle A_{1}:N(0),}

A2:Русский Русский x(N(x)→ → N(x♫)){displaystyle A_{2}:forall x(N(x)to N(x))}}

A3:¬ ¬ consuming consuming x(N(x)∧ ∧ 0=x♫){displaystyle A_{3}:neg exists x(N(x)land 0=x')}}

A4:Русский Русский xРусский Русский and((N(x)∧ ∧ N(and)∧ ∧ x♫=and♫)→ → x=and){displaystyle A_{4}:forall xforall y(N(x)land N(y)land x'=y')to x=y)}

A5:(φ φ (0)∧ ∧ Русский Русский x(φ φ (x)→ → φ φ (x♫)))→ → Русский Русский xφ φ (x){displaystyle A_{5}:{Big (}phi (0)land forall x(phi (x)to phi (x)){Big)}to forall xphi (x)}

A5♫:Русский Русский φ φ (φ φ (0)∧ ∧ Русский Русский x((φ φ (x)→ → φ φ (x♫))→ → Русский Русский xφ φ (x))){displaystyle A_{5}':forall phi {bigg (}phi (0)land forall x{Big (}(phi(x)to phi (x))to forall x phi (x){Big)}{bigg)}}}}

Changing the axioms to include 0 is just a matter of changing every occurrence of 1 to 0. However, the definitions (or axioms) of addition and multiplication require a few more slight adjustments:

Definitions of addition and multiplication:

D1:{displaystyle D_{1}:,}	n+0=n{displaystyle n+0=n,}
	n+m♫=(n+m)♫{displaystyle n+m'=(n+m)',}

D2:{displaystyle D_{2}:,}	n× × 0=0{displaystyle ntimes 0=0,}
	n× × m♫=(n× × m)+n{displaystyle ntimes m'=(ntimes m)+n,}

Axioms of addition and multiplication:

A6:{displaystyle A_{6}:,}	Русский Русский n(n+0=n){displaystyle forall n(n+0=n)}
	Русский Русский nРусский Русский m(n+m♫=(n+m)♫){displaystyle forall nforall m(n+m'=(n+m)'}

A7:{displaystyle A_{7}:,}	Русский Русский n(n× × 0=0){displaystyle forall n(ntimes 0=0),}
	Русский Русский nРусский Русский m(n× × m♫=(n× × m)+n){displaystyle forall nforall m(ntimes m'=(ntimes m)+n),}

Unintentional models

A model is an interpretation of primitive symbols that makes all axioms true. For example, interpreting symbol 0 as number zero, and preaching N{displaystyle {mathcal {N}}} like natural numbers, the first axiom is true, because it is true that “zero is a natural number”. The same applies to all other axioms: under the natural interpretations of 0{displaystyle}, N{displaystyle {mathcal {N}}} and x♫{displaystyle x}each of the axioms is true. Then, the natural interpretations of primitive symbols are a model of the arithmetic of Peano.

Originally, Peano proposed axioms to characterize natural numbers, and primitive symbols should be interpreted in this natural way. However, symbols that designate primitive concepts admit other interpretations, some of which will also be models. For example, the symbol could be interpreted 0{displaystyle} as number two (to simplify explanation we do not understand zero as a pair), a N{displaystyle {mathcal {N}}} as the preached “to be a number pair”, and x♫{displaystyle x} as the successor of the successor, instead of the immediate successor. In such a case, axioms would have to be understood as follows:

1. The two is a pair number.
2. Yeah. n{displaystyle n} is a number pair, then the successor of the successor n{displaystyle n} It's also a number pair.
3. The two is not the successor of any number pair.
4. If there are two pair numbers n{displaystyle n} and m{displaystyle m} with the same successor of successor, then n{displaystyle n} and m{displaystyle m} It's the same number pair.
5. If the two belong to a set, and given a number to any, the successor to that number also belongs to that set, then all the pair numbers belong to that set.

Under this interpretation, all axioms are true, and axioms no longer define natural numbers, but even numbers. It is also possible to find models (i.e. interpretations that make all axioms true) outside mathematics. For example, 0 could be interpreted as the first second after the Big Bang, to N{displaystyle {mathcal {N}}} as the preached “to be a second”, and x♫{displaystyle x} like the second after x{displaystyle x}. Under this interpretation (and assuming time is infinite) axioms are also true.

Models that were not originally planned are called non-intended models, and there are infinitely many unintended models of Peano arithmetic. Strictly speaking, Peano arithmetic does not define the set of natural numbers, but rather the broader notion of a mathematical sequence or arithmetic progression of natural numbers.