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 is In fact this holds for all ordinals We write as 1 The set of all finite ordinals is transitive and hence itself an ordinal It is the least infinite ordinal and is denoted It is the canonical example of a countable set and its cardinality is denoted Like the finite ordinals has a successor 1 namely 0 1 2 That in turn has a successor 0 1 2 1 and we can continue in this way until we reach 0 1 2 1 2 denoted 2 Continuing faster we eventually reach 2 3 We could consider 3 1 but moving yet faster we arrive at 4 5 and eventually denoted 2 Picking up yet more speed we arrive at 3 4 and then Getting into high gear we speed past to arrive at 0 the least ordinal satisfying Throughout this entire sequence every ordinal has been countable that is its cardinality has been However one can not infer from this that every countable ordinal is 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 Instead it is order isomorphic to 2 because the set of even numbers standardly ordered is order isomorphic to and likewise the odd numbers Furthermore we have not come close to exhausting the countable ordinals 0 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 1 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 1 2 3 with corresponding cardinalities The cardinality of an ordinal defines a many to one association from ordinals to cardinals with all ordinals between i inclusive and i 1 exclusive having the same cardinality