Definition:Cardinality of Finite Class

Definition

Let $A$ be a class.

Definition 1

Let $A$ be such that:

there exists a bijection $\phi$ from $A$ to $n$

where $n$ is a natural number as defined by the von Neumann construction.

Then $A$ has cardinality $n$.

Definition 2

Let $A$ be such that:

there exists a bijection $\phi$ from $A$ to the set $\set {1, 2, \dotsc, n} = n^+ \setminus \set 0$

where:

$n$ is a natural number as defined by the von Neumann construction

$n^+$ is the successor of $n$.

Then $A$ has cardinality $n$.

Also known as

This condition can more obviously be put as:

$A$ has $n$ elements.

Also see

• Equivalence of Definitions of Cardinality of Finite Class

• Finite Class is Set