Editing Aleph number (section)

==Aleph-one==
{{Redirect|Aleph One}}
<math>\aleph_1</math> is the cardinality of the set of all countable [[ordinal number]]s.<ref>{{Cite web |title=Power of the continuum {{!}} mathematics {{!}} Britannica |url=https://www.britannica.com/science/power-of-the-continuum |access-date=2025-02-06 |website=www.britannica.com |language=en}}</ref> This set is denoted by <math>\omega_1</math> (or sometimes Ω). The set <math>\omega_1</math> is itself an ordinal number larger than all countable ones, so it is an [[uncountable set]]. Therefore,  <math>\aleph_1</math> is the smallest cardinality that is larger than <math>\aleph_0,</math> the smallest infinite cardinality.

The definition of <math>\aleph_1</math> implies (in ZF, [[Zermelo–Fraenkel set theory]] ''without'' the axiom of choice) that no cardinal number is between <math>\aleph_0</math> and <math>\aleph_1.</math> If the [[axiom of choice]] is used, it can be further proved that the class of cardinal numbers is [[totally ordered]], and thus <math>\aleph_1</math> is the second-smallest infinite cardinal number. One can show one of the most useful properties of the set {{tmath|\omega_1}}: Any countable subset of <math>\omega_1</math> has an upper bound in <math>\omega_1</math> (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 <math>\aleph_0</math>: Every finite set of natural numbers has a maximum which is also a natural number, and [[finite unions]] of finite sets are finite.

An example application of the ordinal <math>\omega_1</math> 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 space]]s, [[group theory|group]]s, 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 <math>\omega_1.</math>