User blog comment:Deedlit11/Ordinal Notations VI: Up to a weakly compact cardinal/@comment-36527-20130718191706/@comment-5529393-20130719035037

Yup, an ordinal is a cardinal if it is the smallest ordinal of a given cardinality (so for example omega is the smallest infinite ordinal, Omega_1 is the smallest uncountable ordinal, etc.). We "collapse" these cardinals to ultimately define large countable ordinals.