User blog comment:Billicusp/messing around/@comment-27513631-20160501153403

If you want to go away from naïve extensions, then you don't want to add recursion on top of nth order set theory, you want some mechanism of diagonalising across it. The simplest way to do this would be either to index the theories with arbitrary ordinals, (I don't know if that's possible when you get down to the semantics), or, failing that, to augment 1st order set theory with the symbol A, such that Ak = A(n,n) when k in a natural number, and use that system.