# Cardinality/Examples/3

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## Example of Cardinality

Let $S$ be a set.

Then $S$ has cardinality $3$ if and only if:

\(\ds \exists x: \exists y: \exists z:\) | \(\) | \(\ds x \in S \land y \in S \land z \in S\) | ||||||||||||

\(\ds \) | \(\land\) | \(\ds x \ne y \land x \ne z \land y \ne z\) | ||||||||||||

\(\ds \) | \(\land\) | \(\ds \forall w: \paren {w \in S \implies \paren {w = x \lor w = y \lor w = z} }\) |

That is:

- $S$ contains elements which can be labelled $x$, $y$ and $z$
- Each of these elements is distinct from the others
- Every element of $S$ is either $x$, $y$ or $z$.

## Historical Note

Jules Henri PoincarĂ© and others questioned the validity and usefulness of this definition of **Cardinality $3$**, in that one needs to have counted the elements of $S$ before starting.

Hence the intuitive notion of $3$-ness is argued to be prior to this definition.

## Sources

- 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 1$ Preliminaries