Editing Well-order (section)

== Ordinal numbers ==
{{main|Ordinal number}}

Every well-ordered set is uniquely [[order isomorphic]] to a unique [[ordinal number]], called the [[order type]] of the well-ordered set. The position of each element within the ordered set is also given by an ordinal number. In the case of a finite set, the basic operation of [[counting]], to find the ordinal number of a particular object, or to find the object with a particular ordinal number, corresponds to assigning ordinal numbers one by one to the objects. The size (number of elements, [[cardinal number]]) of a finite set is equal to the order type.<ref>{{cite journal
 | last1 = Bonnet | first1 = Rémi
 | last2 = Finkel | first2 = Alain
 | last3 = Haddad | first3 = Serge
 | last4 = Rosa-Velardo | first4 = Fernando
 | doi = 10.1016/j.ic.2012.11.003
 | journal = Information and Computation
 | mr = 3016456
 | pages = 1–22
 | title = Ordinal theory for expressiveness of well-structured transition systems
 | volume = 224
 | year = 2013}}</ref> Counting in the everyday sense typically starts from one, so it assigns to each object the size of the initial segment with that object as last element. Note that these numbers are one more than the formal ordinal numbers according to the isomorphic order, because these are equal to the number of earlier objects (which corresponds to counting from zero). Thus for finite {{mvar|n}}, the expression "{{mvar|n}}-th element" of a well-ordered set requires context to know whether this counts from zero or one. In an expression "{{mvar|β}}-th element" where {{mvar|β}} can also be an infinite ordinal, it will typically count from zero.

For an infinite set, the order type determines the [[cardinality]], but not conversely: sets of a particular infinite cardinality can have well-orders of many different types (see {{slink||Natural numbers}}, below, for an example). For a [[countably infinite]] set, the set of possible order types is uncountable.