Equivalence of Definitions of Cardinality of Finite Class

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Theorem

The following definitions of the concept of Cardinality of Finite Class are equivalent:

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 $n$.


Proof

Let $A_1$ be the class which has a bijection $\phi_1$ from $A_1$ to $n$.

Let $A_2$ be the class which has a bijection $\phi_2$ from $A_2$ to $n^+ \setminus \set 0$.

Consider the mapping $\phi: A_1 \to A_2$ defined as:

$\forall k \in n: \map {\phi_1} k = k^+$

$\phi$ is trivially a bijection.

The result follows.

$\blacksquare$


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