Wait wait. Transfinite induction uses ordinals ( https://en.wikipedia.org/wiki/Large_countable_ordinal https://en.wikipedia.org/wiki/Large_countable_ordinal )

Wait wait. Transfinite induction uses ordinals ( https://en.wikipedia.org/wiki/Large_countable_ordinal https://en.wikipedia.org/wiki/Large_countable_ordinal ) To get large cardinals (beyond ZFC) you need large cardinal axioms. ( https://en.wikipedia.org/wiki/List_of_large_cardinal_properties https://en.wikipedia.org/wiki/List_of_large_cardinal_propert... )