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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 Poincaré 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.