Editing Natural number (section)

===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 although Levy attributes the idea to unpublished work of Zermelo in 1916.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.