Cardinal number

https://en.wikipedia.org/wiki/Cardinal_number

Apart from the functional definition (given above), another way to define cardinality is to define it as a specific object.

The relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the class of all sets.

The equivalence class of a set A under this relation then consists of all those sets which have the same cardinality as A.

There are two ways to define the cardinality of a set:

• The cardinality of a set A is defined as its equivalence class under equinumerosity.

• A representative set is designated for each equivalence class. The most common choice is the initial ordinal in that class. This is usually taken as the definition of cardinal number in axiomatic set theory.

Assuming axiom of choice (AC), the cardinalities of the infinite sets are denoted $\aleph_{0} \lt \aleph_{1} \lt \aleph_{2} \lt \ldots$

For each ordinal $\alpha, \aleph_{\alpha+1}$ is the least cardinal number greater than $\aleph_{\alpha}$.