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 card("S") to be the set of sets with the same cardinality as "S" of minimum possible rank. This has the property that card("S") = card("T") if and only if "S" and "T" have the same cardinality. (The set card("S") does not have the same cardinality of "S" in general, but all its elements do.)