Existence of Hartogs Number

Theorem

Let $S$ be a set.

Then $S$ has a Hartogs number.

Proof 1

From Hartogs' lemma there exists an ordinal $\alpha$ such that there is no injection from $\alpha$ to $S$.

We also have that Ordinals are Well-Ordered.

It follows from the definition of well-ordering that there exists a smallest such ordinal.

Hence the result.

$\blacksquare$

Proof 2

Follows immediately from Cardinal Equal to Collection of All Dominated Ordinals.

The collection of all dominated ordinals:

$\set {y \in \On: y \preccurlyeq S}$

is the Hartogs number of $S$.

$\blacksquare$