In mathematics, an ordinal number, or just ordinal, is a transitive set of ordinals, or hereditarily transitive set.  That is, every element of an ordinal is transitive and its elements in turn are transitive and so on down.  The Axiom of regularity stops this recursion after finitely many steps, namely at the empty set.

The finite ordinals are 0 = {}, 1 = {0}, 2 = {0,1}, 3 = {0,1,2}, ….  This is a consequence of two remarkable facts, that every element of a finite transitive set is itself a finite transitive set (so we can drop "ordinal" as a condition for membership in a finite ordinal), and only one set of each finite cardinality can be transitive, whence the finite ordinals can be identified with their cardinalities as per the foregoing enumeration.

It is evident for the finite ordinals that the successor of an ordinal α is α∪{α}.  In fact this holds for all ordinals.  We write α∪{α} as α+1.

The set of all finite ordinals is transitive and hence itself an ordinal.  It is the least infinite ordinal, and is denoted ω.  It is the canonical example of a countable set, and its cardinality is denoted .

Like the finite ordinals, ω has a successor ω+1, namely {0,1,2,…,ω}.  That in turn has a successor {0,1,2,…,ω,ω+1}, and we can continue in this way until we reach {0,1,2,…,ω,ω+1,ω+2…} = ω+ω, denoted ω·2.  Continuing faster we eventually reach ω·2+ω = ω·3.  We could consider ω·3+1, but moving yet faster we arrive at ω·4, ω·5, and eventually ω·ω, denoted ω2.  Picking up yet more speed, we arrive at ω3, ω4, … and then ωω.  Getting into high gear we speed past ωω, ωωω, ωωωω, …, to arrive at ε0, the least ordinal α satisfying α = ωα.

Throughout this entire sequence, every ordinal has been countable, that is, its cardinality has been .  However one cannot infer from this that every countable ordinal is ω.  To see this, consider listing all infinitely many even numbers first, and then continue by listing the odd numbers.  Then 1 is preceded only by even numbers, but there is no even number immediately before 1 because there is no largest even number.  So unlike the usual enumeration of the natural numbers, in which every number but 0 has a predecessor, this enumeration is structurally different, that is, it is not order isomorphic to ω.  Instead it is order isomorphic to ω+ω = ω·2, because the set of even numbers standardly ordered is order isomorphic to ω, and likewise the odd numbers.

Furthermore we have not come close to exhausting the countable ordinals: ε0+1 is countable, and so on.  To get past the countable ordinals requires a new insight.
  
Every ordinal is linearly ordered by inclusion, for example 2 < 4 because {0,1} is a subset of {0,1,2,3}.  This linear order is in fact a well-order, that is, every nonempty set of ordinals has a least element.  Hence for every predicate that is not false of all ordinals, there exists a least ordinal satisfying that predicate.  Furthermore no two ordinals are order isomorphic, and every well-ordered set is order isomorphic to some ordinal, whence the ordinals can serve to encode the order types of all well-ordered sets however large.

There exist uncountable ordinals, whence by the foregoing there exists a least uncountable ordinal, denoted ω1.  The cardinality of this ordinal is denoted , which is the next cardinal after .  There is a sequence of increasingly large ordinals as measured by their cardinality, denoted ω1, ω2, ω3, …, with corresponding cardinalities , , , ….  The cardinality of an ordinal defines a many to one association from ordinals to cardinals, with all ordinals between ωi inclusive and ωi+1 exclusive having the same cardinality .