stylistic:tone, structural:summarization
Aleph-one

ℵ is, by definition, the cardinality of the set of all countable ordinal numbers. This set is denoted by ω (or sometimes Ω). The set ω is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore, ℵ is distinct from ℵ. The definition of ℵ implies (in ZF, Zermelo–Fraenkel set theory without the axiom of choice) that no cardinal number is between ℵ and ℵ. If the axiom of choice is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus ℵ is the second-smallest infinite cardinal number. One can show one of the most useful properties of the set ω: Any countable subset of ω has an upper bound in ω (this follows from the fact that the union of a countable number of countable sets is itself countable). This fact is analogous to the situation in ℵ: Every finite set of natural numbers has a maximum which is also a natural number, and finite unions of finite sets are finite.

The ordinal ω is actually a useful concept, if somewhat exotic-sounding. An example application is "closing" with respect to countable operations; e.g., trying to explicitly describe the σ-algebra generated by an arbitrary collection of subsets (see e.g. Borel hierarchy). This is harder than most explicit descriptions of "generation" in algebra (vector spaces, groups, etc.) because in those cases we only have to close with respect to finite operations – sums, products, etc. The process involves defining, for each countable ordinal, via transfinite induction, a set by "throwing in" all possible countable unions and complements, and taking the union of all that over all of ω.