Editing Natural number (section)

==Generalizations==
Two important generalizations of natural numbers arise from the two uses of counting and ordering: [[cardinal number]]s and [[ordinal number]]s.
* A natural number can be used to express the size of a finite set; more precisely, a cardinal number is a measure for the size of a set, which is even suitable for infinite sets. The numbering of cardinals usually begins at zero, to accommodate the [[empty set]] <math>\emptyset</math>. This concept of "size" relies on maps between sets, such that two sets have [[equinumerosity|the same size]], exactly if there exists a [[bijection]] between them. The set of natural numbers itself, and any bijective image of it, is said to be ''[[countable set|countably infinite]]'' and to have [[cardinality]] [[Aleph number#Aleph-null|aleph-null]] ({{math|{{not a typo|ℵ}}<sub>0</sub>}}).
* Natural numbers are also used as [[Ordinal numbers (linguistics)|linguistic ordinal numbers]]: "first", "second", "third", and so forth. The numbering of ordinals usually begins at zero, to accommodate the order type of the  [[empty set]] <math>\emptyset</math>. This way they can be assigned to the elements of a totally ordered finite set, and also to the elements of any [[well-order]]ed countably infinite set without [[limit points]]. This assignment can be generalized to general well-orderings with a cardinality beyond countability, to yield the ordinal numbers. An ordinal number may also be used to describe the notion of "size" for a well-ordered set, in a sense different from cardinality: if there is an [[order isomorphism]] (more than a bijection) between two well-ordered sets, they have the same ordinal number. The first ordinal number that is not a natural number is expressed as {{math|ω}}; this is also the ordinal number of the set of natural numbers itself.

The least ordinal of cardinality {{math|{{not a typo|ℵ}}<sub>0</sub>}} (that is, the [[Von Neumann cardinal assignment|initial ordinal]] of {{math|{{not a typo|ℵ}}<sub>0</sub>}}) is {{math|ω}} but many well-ordered sets with cardinal number {{math|{{not a typo|ℵ}}<sub>0</sub>}} have an ordinal number greater than {{math|ω}}.

For [[finite set|finite]] well-ordered sets, there is a one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by the same natural number, the number of elements of the set. This number can also be used to describe the position of an element in a larger finite, or an infinite, [[sequence]].

A countable [[non-standard model of arithmetic]] satisfying the Peano Arithmetic (that is, the first-order Peano axioms) was developed by [[Skolem]] in 1933. The [[hypernatural]] numbers are an uncountable model that can be constructed from the ordinary natural numbers via the [[ultrapower construction]]. Other generalizations are discussed in {{section link|Number#Extensions of the concept}}.

[[Georges Reeb]] used to claim provocatively that "The naïve integers don't fill up <math>\mathbb{N}</math>".<ref>{{cite journal |title=Approaches To Analysis With Infinitesimals Following Robinson, Nelson, And Others |journal=Real Analysis Exchange |date=2017 |volume=42 |issue=2 |pages=193–253 |doi=10.14321/realanalexch.42.2.0193|doi-access=free|arxiv=1703.00425 |last1=Fletcher |first1=Peter |last2=Hrbacek |first2=Karel |last3=Kanovei |first3=Vladimir |last4=Katz |first4=Mikhail G. |last5=Lobry |first5=Claude |last6=Sanders |first6=Sam }}</ref>