Cardinality - Cardinal Numbers

Cardinal Numbers

Above, "cardinality" was defined functionally. That is, the "cardinality" of a set was not defined as a specific object itself. However, such an object can be defined as follows.

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":

1. The cardinality of a set A is defined as its equivalence class under equinumerosity.
2. 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.

The cardinalities of the infinite sets are denoted

For each ordinal α, ℵ_{α + 1} is the least cardinal number greater than ℵ_α.

The cardinality of the natural numbers is denoted aleph-null (ℵ₀), while the cardinality of the real numbers is denoted by c, and is also referred to as the cardinality of the continuum. Cantor showed, using the diagonal argument, that c>ℵ₀. We can show that c = 2ℵ₀; this also being the cardinality of the set of all subsets of the natural numbers. The continuum hypothesis says that ℵ₁ = 2ℵ₀, i.e. 2ℵ₀ is the smallest cardinal number bigger than ℵ₀, i.e. there is no set whose cardinality is strictly between that of the integers and that of the real numbers. The continuum hypothesis still remains unresolved in an "absolute" sense. See below for more details on the cardinality of the continuum.