Sorry, classes of well-orderings of subsets of the natural numbers up to order isomorphism: https://en.wikipedia.org/wiki/Hartogs_number https://en.wikipedia.or

Sorry, classes of well-orderings of subsets of the natural numbers up to order isomorphism: https://en.wikipedia.org/wiki/Hartogs_number https://en.wikipedia.org/wiki/Hartogs_number. And yes it’s equivalent to your definition due to ordinal magic as far as I can tell (or maybe because cardinals are by definition well ordered, I’m clearly not an expert).