Editing Aleph number (section)

==Role of axiom of choice==
The cardinality of any infinite [[ordinal number]] is an aleph number. Every aleph is the cardinality of some ordinal. The least of these is its [[initial ordinal]]. Any set whose cardinality is an aleph is [[equinumerous]] with an ordinal and is thus [[well-order]]able.

Each [[finite set]] is well-orderable, but does not have an aleph as its cardinality.

Over ZF, the assumption that the cardinality of each [[infinite set]] is an aleph number is equivalent to the existence of a well-ordering of every set, which in turn is equivalent to the [[axiom of choice]]. ZFC set theory, which includes the axiom of choice, implies that every infinite set has an aleph number as its cardinality (i.e. is equinumerous with its initial ordinal), and thus the initial ordinals of the aleph numbers serve as a class of representatives for all possible infinite cardinal numbers.

When cardinality is studied in ZF without the axiom of choice, it is no longer possible to prove that each infinite set has some aleph number as its cardinality; the sets whose cardinality is an aleph number are exactly the infinite sets that can be well-ordered. The method of [[Scott's trick]] is sometimes used as an alternative way to construct representatives for cardinal numbers in the setting of ZF. For example, one can define <math>\text{card}(S)</math> to be the set of sets with the same cardinality as <math>S</math> of minimum possible rank. This has the property that <math>\text{card}(S) = \text{card}(T)</math> if and only if <math>S</math> and <math>T</math> have the same cardinality. (The set <math>\text{card}(S)</math> does not have the same cardinality of <math>S</math> in general, but all its elements do.)