User blog comment:JHeroJr/Is there a system which inductively defines some 1st, 2nd, 3rd, 4th, etc. set theories?/@comment-35470197-20190907053135/@comment-35470197-20190908042518

The (proof theoretic) ordinal corresponding to (consistent recursively axiomised) set theory is always recursive, and hence there is no system of higher order set theories generating all countable ordinals in that way.