Hartogs' Lemma

For any set , there’s an ordinal that doesn’t inject into .

Proof

This is a generalization of Uncountable ordinal.
We form the set
consisting of Order Types of well-orderings of subsets of .
Let
If injects into ,
then induces a well-ordering on a subset of of order-type .
Thus ,
so
which is a contradiction

Notation

The least ordinal that doesn’t inject into is denoted .