formula_3 is the cardinality of the set of all countable ordinal numbers, called formula_24 or sometimes formula_25. This formula_24 is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore, formula_3 is distinct from formula_2. The definition of formula_3 implies (in ZF, Zermelo–Fraenkel set theory "without" the axiom of choice) that no cardinal number is between formula_2 and formula_3. If the axiom of choice is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus formula_3 is the second-smallest infinite cardinal number. Using the axiom of choice, one can show one of the most useful properties of the set formula_24: any countable subset of formula_24 has an upper bound in formula_35 (This follows from the fact that the union of a countable number of countable sets is itself countable—one of the most common applications of the axiom of choice.) This fact is analogous to the situation in formula_2: every finite set of natural numbers has a maximum which is also a natural number, and finite unions of finite sets are finite.