Cardinal Zero is Less than Cardinal One
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The zero cardinal $0$ is less than one:
- $0 < 1$
We have that the Cardinals are Totally Ordered.
Let $\RR \subseteq \O \times \set \O$ be any arbitrary relation between $\O$ and $\set \O$.
We have that $\RR$ is vacuously many-to-one.
Also vacuously, $\RR$ is left-total.
Thus by definition, $\RR$ is in fact a mapping.
From Empty Mapping is Unique, this relation $\RR$ is the unique mapping from $\O$ to $\set \O$.
Also vacuously, $\RR$ is an injection.
So, by definition, $\set \O$ dominates $\O$.
Since Empty Set is Subset of All Sets, we have $\O \subseteq \set \O$.
Hence the result.
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 8$