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 .