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:
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.
$\blacksquare$
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 6$ Finite Sets