Editing Natural number (section)

==Formal definitions==

There are two standard methods for formally defining natural numbers. The first one, named for [[Giuseppe Peano]], consists of an autonomous [[axiomatic theory]] called [[Peano arithmetic]], based on few axioms called [[Peano axioms]].

The second definition is based on [[set theory]]. It defines the natural numbers as specific [[set (mathematics)|set]]s. More precisely, each natural number {{mvar|n}} is defined as an explicitly defined set, whose elements allow counting the elements of other sets, in the sense that the sentence "a set {{mvar|S}} has {{mvar|n}} elements" means that there exists a [[one to one correspondence]] between the two sets {{mvar|n}} and {{mvar|S}}.

The sets used to define natural numbers satisfy Peano axioms. It follows that every [[theorem]] that can be stated and proved in Peano arithmetic can also be proved in set theory. However, the two definitions are not equivalent, as there are theorems that can be stated in terms of Peano arithmetic and proved in set theory, which are not ''provable'' inside Peano arithmetic. A probable example is [[Fermat's Last Theorem]].

The definition of the integers as sets satisfying Peano axioms provide a [[model (mathematical logic)|model]] of Peano arithmetic inside set theory. An important consequence is that, if set theory is [[consistent]] (as it is usually guessed), then Peano arithmetic is consistent. In other words, if a contradiction could be proved in Peano arithmetic, then set theory would be contradictory, and every theorem of set theory would be both true and wrong. 

===Peano axioms===
{{Main|Peano axioms}}

The five Peano axioms are the following:<ref>{{cite encyclopedia
 |editor-first=G.E. |editor-last=Mints
 |title=Peano axioms
 |encyclopedia=Encyclopedia of Mathematics
 |publisher=[[Springer Science+Business Media|Springer]], in cooperation with the [[European Mathematical Society]]
 |url=http://www.encyclopediaofmath.org/index.php/Peano_axioms
 |url-status=live |access-date=8 October 2014
|archive-url=https://web.archive.org/web/20141013163028/http://www.encyclopediaofmath.org/index.php/Peano_axioms
 |archive-date=13 October 2014
}}</ref>{{efn|{{harvtxt|Hamilton|1988|pages=117&nbsp;ff}} calls them "Peano's Postulates" and begins with "1.{{spaces|2}}0 is a natural number."<br/>
{{harvtxt|Halmos|1960|page=46}} uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I){{spaces|2}}{{math|0 ∈ ω}} (where, of course, {{math|0 {{=}} ∅}}" ({{math|ω}} is the set of all natural numbers).<br/>
{{harvtxt|Morash|1991}} gives "a two-part axiom" in which the natural numbers begin with 1. (Section 10.1: ''An Axiomatization for the System of Positive Integers'')
}}

# 0 is a natural number.
# Every natural number has a successor which is also a natural number.
# 0 is not the successor of any natural number.
# If the successor of <math> x </math> equals the successor of <math> y </math>, then <math> x</math> equals <math> y</math>.
# The [[axiom of induction]]: If a statement is true of 0, and if the truth of that statement for a number implies its truth for the successor of that number, then the statement is true for every natural number.

These are not the original axioms published by Peano, but are named in his honor. Some forms of the Peano axioms have 1 in place of 0. In ordinary arithmetic, the successor of <math> x</math> is <math> x + 1</math>.

===Set-theoretic definition===
{{Main|Set-theoretic definition of natural numbers}}

Intuitively, the natural number {{mvar|n}} is the common property of all [[set (mathematics)|set]]s that have {{mvar|n}} elements. So, it seems natural to define {{mvar|n}} as an [[equivalence class]] under the relation "can be made in [[one to one correspondence]]". This does not work in all [[set theory|set theories]], as such an equivalence class would not be a set{{efn|In some set theories, e.g., [[New Foundations]], a [[universal set]] exists and Russel's paradox cannot be formulated.}} (because of [[Russell's paradox]]). The standard solution is to define a particular set with {{mvar|n}} elements that will be called the natural number {{mvar|n}}.

The following definition was first published by [[John von Neumann]],<ref name="vonNeumann1923pp199-208">{{Harvp|von&nbsp;Neumann|1923}}</ref> although Levy attributes the idea to unpublished work of Zermelo in 1916.<ref name="Levy">{{harvp|Levy|1979|page=52}}</ref> As this definition extends to [[infinite set]] as a definition of [[ordinal number]], the sets considered below are sometimes called [[von Neumann ordinals]].

The definition proceeds as follows:
* Call {{math|0 {{=}} {{mset| }}}}, the [[empty set]].
* Define the ''successor'' {{math|''S''(''a'')}} of any set {{mvar|a}} by {{math|''S''(''a'') {{=}} ''a'' ∪ {{mset|''a''}}}}.
* By the [[axiom of infinity]], there exist sets which contain 0 and are [[closure (mathematics)|closed]] under the successor function. Such sets are said to be ''inductive''. The intersection of all inductive sets is still an inductive set.
* This intersection is the set of the ''natural numbers''.

It follows that the natural numbers are defined iteratively as follows:
:*{{math|0 {{=}} {{mset| }}}},
:*{{math|1 {{=}} 0 ∪ {{mset|0}} {{=}} {{mset|0}} {{=}} {{mset|{{mset| }}}}}},
:*{{math|2 {{=}} 1 ∪ {{mset|1}} {{=}} {{mset|0, 1}} {{=}} {{mset|{{mset| }}, {{mset|{{mset| }}}}}}}},
:*{{math|3 {{=}} 2 ∪ {{mset|2}} {{=}} {{mset|0, 1, 2}}}} {{math|{{=}} {{mset|{{mset| }}, {{mset|{{mset| }}}}, {{mset|{{mset| }}, {{mset|{{mset| }}}}}}}}}},
:*{{math|''n'' {{=}} ''n''−1 ∪ {{mset|''n''−1}} {{=}} {{mset|0, 1, ..., ''n''−1}}}} {{math|{{=}} {{mset|{{mset| }}, {{mset|{{mset| }}}}, ..., {{mset|{{mset| }}, {{mset|{{mset| }}}}, ...}}}}}}, 
:* etc.

It can be checked that the natural numbers satisfy the [[Peano axioms]].

With this definition, given a natural number {{math|''n''}}, the sentence "a set {{mvar|S}} has {{mvar|n}} elements" can be formally defined as "there exists a [[bijection]] from {{mvar|n}} to {{mvar|S}}." This formalizes the operation of ''counting'' the elements of {{mvar|S}}. Also, {{math|''n'' ≤ ''m''}} if and only if {{math|''n''}} is a [[subset]] of {{math|''m''}}. In other words, the [[set inclusion]] defines the usual [[total order]] on the natural numbers. This order is a [[well-order]].

It follows from the definition that each natural number is equal to the set of all natural numbers less than it. This definition, can be extended to the [[von Neumann ordinal|von Neumann definition of ordinals]] for defining all [[ordinal number]]s, including the infinite ones: "each ordinal is the well-ordered set of all smaller ordinals."

If one [[finitism|does not accept the axiom of infinity]], the natural numbers may not form a set. Nevertheless, the natural numbers can still be individually defined as above, and they still satisfy the Peano axioms.

There are other set theoretical constructions. In particular, [[Ernst Zermelo]] provided a construction that is nowadays only of historical interest, and is sometimes referred to as '''{{vanchor|Zermelo ordinals}}'''.<ref name="Levy"/> It consists in defining {{math|0}} as the empty set, and {{math|''S''(''a'') {{=}} {{mset|''a''}}}}.

With this definition each nonzero natural number is a [[singleton set]]. So, the property of the natural numbers to represent [[cardinalities]] is not directly accessible; only the ordinal property (being the {{mvar|n}}th element of a sequence) is immediate. Unlike von Neumann's construction, the Zermelo ordinals do not extend to infinite ordinals.