Aleph Number - Aleph-one

Aleph-one

is the cardinality of the set of all countable ordinal numbers, called ω₁ or (sometimes) Ω. This ω₁ is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore is distinct from . The definition of implies (in ZF, Zermelo–Fraenkel set theory without the axiom of choice) that no cardinal number is between and . If the axiom of choice (AC) is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus is the second-smallest infinite cardinal number. Using AC we can show one of the most useful properties of the set ω₁: any countable subset of ω₁ has an upper bound in ω₁. (This follows from the fact that a countable union of countable sets is countable, one of the most common applications of AC.) This fact is analogous to the situation in : every finite set of natural numbers has a maximum which is also a natural number; that is, finite unions of finite sets are finite.

ω₁ 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, and the like. 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 ω₁.