Cardinal Zero is Less than Cardinal One

Theorem

The zero cardinal $0$ is less than one:

$0 < 1$

Proof

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.

$\blacksquare$

Sources

• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 8$