Axioms of Peano

The axioms of Peano are, in Mathématiques, a whole of Axiome S of second order proposed by Giuseppe Peano to define the Arithmétique.

Axioms

The axiomatic definition of the natural whole of Peano is usually described informellement by five axioms:

1. the element called zero and noted: $0$, is a natural entirety.

2. Entire naturalness $n$ has a single successor, noted $s\; (N)$ or $Sn$.
3. No entirety natural has $0$ for successor.
4. Two natural entireties having even successor are equal.
5. If a whole of natural entireties contains $0$ and contains the successor of each one of his elements, then this unit is equal to $\backslash \; mathbb\; \{NR\}$.

The first axiom makes it possible to pose that the whole of the natural entireties is not Vide, the third which it has a first element and the fifth that it checks the principle of recurrence.

In a more formal way, the triplet $\backslash \; left\; (E,\; X,\; S\; \backslash \; right)$ satisfies the following properties:

1. $E$ is a Ensemble, $x$ is an element of $E$, $s$ is a application of $E$ in itself.

2. $x\; \backslash \; notin\; S\; \backslash \; left\; (E\; \backslash \; right)$
3. $s$ is injective
4. All Sous-ensemble $F$ of $E$ container $x$ and stable by $s$ (i.e. $s\; \backslash \; left\; (F\; \backslash \; right)\; \backslash \; subset\; F$) is equal to $E$.

Such a structure is called structure of Dedekind-Peano (according to the mathematician Richard Dedekind)

Arithmetic of Peano

The arithmetic one of Peano is the restriction of the axioms of Peano on the language of arithmetic of the first order $\backslash \; \{0,\; S,\; +,\; \backslash \; cdot\; \backslash \}$. The variables of the language indicate objects of the field of interpretation, here of the entireties. In this first order language, one does not have variables for the whole of entireties, and one cannot quantify on these units. One cannot thus directly express the recurrence by a statement such as that of the preceding paragraph (“any subset…”). It is considered whereas a subset of $\backslash \; mathbb\; N$ is expressed by a property of its elements, property which one writes in the language of the arithmetic one.

The axioms of Peano then become the 7 following axioms, to which is added, for the recurrence, a diagram of axioms , which represents a countable infinity of axioms (an axiom for each formula of the language):

1. $\backslash \; forall\; X\; \backslash \; lnot\; (sx\; =\; 0)$

2. $\backslash \; forall\; X\; \backslash \; exists\; there\; (\backslash \; lnot\; x=0\; \backslash \; rightarrow\; sy=x)$
3. $\backslash \; forall\; X\; \backslash \; forall\; there\; (sx=sy\; \backslash \; rightarrow\; x=y)$
4. $\backslash \; forall\; X\; (x+0=x)$
5. $\backslash \; forall\; X\; \backslash \; forall\; there\; (x+sy\; =\; S\; (x+y))$
6. $\backslash \; forall\; X\; (X\; \backslash \; cdot\; 0=0)$
7. $\backslash \; forall\; X\; \backslash \; forall\; there\; (X\; \backslash \; cdot\; Sy\; =\; (X\; \backslash \; cdot\; there)\; +\; X)$
8. For any formula $\backslash \; phi\; (X,\; x\_1,\; \backslash \; ldots,\; x\_n)$ with $n+1$ variable free, $\backslash \; forall\; x\_1\; \backslash \; ldots\; \backslash \; forall\; x\_n\; \backslash \; left\; (\backslash \; left\; (\backslash \; phi\; \backslash \; left\; (0,\; x\_1,\; \backslash \; ldots,\; x\_n\; \backslash \; right)\; \backslash \; wedge\; \backslash \; forall\; X\; \backslash \; left\; (\backslash \; phi\; \backslash \; left\; (X,\; x\_1,\; \backslash \; ldots,\; x\_n\; \backslash \; right)\; \backslash \; rightarrow\; \backslash \; phi\; \backslash left\; (Sx,\; x\_1,\; \backslash \; ldots,\; x\_n\; \backslash \; right)\; \backslash \; right)\; \backslash \; right)\; \backslash \; rightarrow\; \backslash \; forall\; X\; \backslash \; phi\; \backslash \; left\; (X,\; x\_1,\; \backslash \; ldots,\; x\_n\; \backslash \; right)\; \backslash \; right)$

The diagram of axioms expresses the recurrence well: in the formula $\backslash \; phi\; (X,\; x\_1,\; \backslash \; ldots,\; x\_n)$, the variables $(x\_1,\; \backslash \; ldots,\; x\_n)$ is parameters, which one can replace by arbitrary entireties $(p\_1,\; \backslash \; ldots,\; p\_n)$. The axiom for the formula $\backslash \; phi\; (X,\; x\_1,\; \backslash \; ldots,\; x\_n)$ becomes, applied to $(p\_1,\; \backslash \; ldots,\; p\_n)$:

$\backslash \; left\; (\backslash \; left\; (\backslash \; phi\; \backslash \; left\; (0,\; p\_1,\; \backslash \; ldots,\; p\_n\; \backslash \; right)\; \backslash \; wedge\; \backslash \; forall\; X\; \backslash \; left\; (\backslash \; phi\; \backslash \; left\; (X,\; p\_1,\; \backslash \; ldots,\; p\_n\; \backslash \; right)\; \backslash \; rightarrow\; \backslash \; phi\; \backslash \; left\; (Sx,\; p\_1,\; \backslash \; ldots,\; p\_n\; \backslash \; right)\; \backslash \; right)\; \backslash \; right)\; \backslash \; rightarrow\; \backslash \; forall\; X\; \backslash \; phi\; \backslash \; left\; (X,\; p\_1,\; \backslash \; ldots,\; p\_n\; \backslash \; right)\; \backslash \; right)$

What expresses although, if the unit $\backslash \; left\; \backslash \; \{X\; \backslash \; in\; \backslash \; mathbb\; NR\; \backslash \; mid\; \backslash \; phi\; \backslash \; left\; (X,\; p\_1,\; \backslash \; ldots,\; p\_n\; \backslash \; right)\; \backslash \; right\; \backslash \}$ contains $0$, and if it contains the successor of each one of his elements, it is $\backslash \; mathbb\; N$.

However, the diagram of axioms does not give any more this property but for the subsets of $\backslash \; mathbb\; N$ which are defined in the language of arithmetic first order: a countable infinity of subsets of $\backslash \; mathbb\; N$.

One can show that the arithmetic one of Peano cannot be finiment axiomatized, unless modifying the language. That does not have thus inevitably great direction to seek to minimize the axioms. One can notice all the same that axiom 2 could be eliminated. It is shown by recurrence, a rather singular recurrence, since it is necessary well to distinguish case 0 from the case successor, but that in this last case, the assumption of recurrence is not useful.

Existence and unicity

The existence of a structure of Dedekind-Peano can be established by a very usual construction within the framework of the Set theory:

• One poses 0 = ∅.

• One defines the “function” (with the intuitive direction) successor $s$ while posing, for any unit $a$, $s\; (a)=a\; \backslash \; cup\; \backslash \; \{has\; \backslash \}$. It is noticed that for all units has and B :

S ( has ) ≠ 0;     S ( has ) = S ( B ) ⇒ has = B .

• a $A$ unit is known as inductive if it contains 0 is if it is closed by successor, i.e. if $a\; \backslash \; in\; A$, then $s\; (a)\; \backslash \; in\; A$.
• the existence of at least an inductive unit is ensured by the Axiome of infinite the. One defines then the structure $\backslash \; left\; (\backslash \; mathbf\; \{NR\},\; \backslash \; empty,\; S|\_\; \backslash \; mathbf\; \{NR\}\; \backslash \; right)$ : $\backslash \; mathbf\; \{NR\}$ is the intersection of all the inductive units and $s|\_\; \backslash \; mathbf\; \{NR\}$ is the restriction of $s$ on $\backslash \; mathbf\; \{NR\}$. This structure satisfies the above mentioned axioms (inter alia it is well ordered). One can define $\backslash \; mathbf\; \{NR\}$ like the whole of the natural entireties.

This unit is also the whole of the finished ordinal of Von Neumann. This construction of NR is not really canonical, essence is that 0 are never a successor and that the successor is injective (and still, that would be enough that it is on the unit obtained), but it makes it possible to build in a simple way and uniform a unit representing each cardinality finished (the entirety N thus built has, as a unit, for cardinal N ), the axiom of infinite making it possible to prove that they form a unit.

Two structures of Dedekind-Peano $\backslash \; left\; (X,\; X,\; F\; \backslash \; right)$ and $\backslash \; left\; (Y,\; there,\; G\; \backslash \; right)$ are known as isomorphous if there exists a Bijection $\backslash \; phi:\; X\; \backslash \; rightarrow\; Y$ such as $\backslash \; phi\; (X)\; =y$ and $\backslash \; phi\; F\; =\; G\; \backslash \; phi$. One can show that all the structures of Dedekind-Peano are isomorphous.

One often finds the notation $\backslash \; mathbb\; \{NR\}$ for the whole of the natural entireties.

Operations and order

The addition and the multiplication are defined on $\backslash \; mathbf\; \{NR\}$ by the axioms of Peano.

The addition on $\backslash \; mathbf\; \{NR\}$ is recursively defined by posing $a+0=a$ and $a+s\; (b)=s\; (a+b)$ for all $a$ and $b$. $\backslash \; left\; (\backslash \; mathbf\; \{NR\},\; +\; \backslash \; right)$ is thus a commutative Monoïde of neutral element $0$. This monoid can be plunged in a group. More the small group the container is that of the integers.

Since $s\; (0)\; =1$, $s\; (b)=s\; (b+0)\; =b+s\; (0)\; =b+1$. The successor of $b$ is simply $b+1$.

In a similar way, by supposing that the addition was defined, the multiplication on $\backslash \; mathbf\; \{NR\}$ is defined by posing $a\; \backslash \; cdot\; 0=0$ and $a\; \backslash \; cdot\; (b+1)\; =\; (has\; \backslash \; cdot\; b)+a$. $\backslash \; left\; (\backslash \; mathbf\; \{NR\},\; \backslash \; cdot\; \backslash \; right)$ is thus a commutative monoid of neutral element $1$.

It is finally possible to define a total Ordre on $\backslash \; mathbf\; \{NR\}$ while posing that $a\; \backslash \; the\; b$ if there exists a $c$ number such as $a+c=b$. Then $\backslash \; mathbf\; \{NR\}$ provided of this kind is a ordered well: any nonempty whole of natural numbers has a smaller element.

Coherence

Under the terms of the second Theorem of incomplétude of Gödel, the non-contradiction of these axioms between them is not consequence of these only axioms: one cannot prove the coherence of arithmetic in the arithmetic one.

A structure of Dedekind-Peano is a model of these axioms. Construction above thus provides a proof of consistency of axiom relative to a theory in which one can define these structures, and to formalize the proof of correction, for example the axiomatic Théorie of the units of Ernst Zermelo. There exist also evidence of relative Cohérence, in particular that of Gerhard Gentzen which provides a precise measurement of the “force” of the arithmetic one: it is enough to add a principle of induction until the countable Ordinal $\backslash \; epsilon\_0$ to be able to show the coherence of the arithmetic one.

Nonstandard models

A model of arithmetic of Peano which is not a structure of Dedekind-Peano, and is thus not isomorphous with $\backslash \; mathbf\; \{NR\}$ is known as “not standard”.

All models not standard of arithmetic contains the natural entireties, which one calls then, “standard” entireties, and which are the elements of the models that one can indicate by terms of the language, the other elements of the model are then called whole not standard.

More precisely if $\backslash \; mathbf\; \{\}\; \backslash ,$ is a nonstandard model of arithmetic, then it exists a injection $f$ of $\backslash \; mathbf\; \{NR\}$ in $\backslash \; mathbf\; \{\}\; \backslash ,$ such as:

• $f\; (0)\; =\; 0\; \backslash \; quad$
• $\backslash \; forall\; N,\; F\; (S\; (N))\; =\; S\; (F\; (N))$

and the image of F is what one calls the whole of the standard entireties of the model.

It is not possible to distinguish the standard entireties from the nonstandard entireties in the language of the arithmetic one, since if a predicate made it possible to characterize the standard entireties, the diagram of recurrence particularized to this predicate would not be valid. One “thus leaves” arithmetic Peano as soon as one reasons on these concepts in a nonstandard model. But, one can be useful oneself of course owing to the fact that the axioms of Peano remain valid in this model. It is shown for example easily that a nonstandard entirety is necessarily higher than a standard entirety. The totality of the order (defined by the addition, to see above), remains valid. If a nonstandard entirety were smaller than a standard entirety, one would show by injectivity of the successor and recurrence which there exists a nonstandard entirety smaller than 0, and 0 would be a successor. Still more simply, it is shown that there cannot be of more small nonstandard entirety, since entire not no one is a successor.

Existence of the nonstandard models

• the Theorem of compactness and the Théorème of Löwenheim-Skolem ensure that there exist nonstandard countable models of arithmetic of Peano which check the same first order statements exactly as $\backslash \; mathbf\; \{NR\}\; \backslash ,$. Abraham Robinson bases the Analyze nonstandard on a model of arithmetic checking in particular this condition.

• There exist also nonstandard models which check false first order statements in $\backslash \; mathbf\; \{NR\}\; \backslash ,$ (moreover, let us recall it, of all the demonstrable statements in Peano, by definition of the concept of model). A true statement in $\backslash \; mathbf\; \{NR\}\; \backslash ,$ is not demonstrable in the arithmetic one of Peano, if and only if there exists a nonstandard model in which this statement is false. The theorems of incomplétude of Gödel thus have as a consequence the existence of such models (which check a formula expressing that the arithmetic one of Peano is incoherent!). A contrario, one can use such models to show that certain statements are not demonstrable in the arithmetic one of Peano.

See too

Internal bonds

• Arithmetic
• Analysis nonstandard

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