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 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.
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