User blog comment:JHeroJr/Is there a system which inductively defines some 1st, 2nd, 3rd, 4th, etc. set theories?/@comment-35470197-20190907053135/@comment-35470197-20190909151243

> Of course, to do this, you need to use a logic that can be used by an ITTM to go through infinite proofs, which I think are called infinitary logics.

You do not need the infiniteness of proof in order to deal with non-recursive set theory. It is completely irrelevant.

> You can inductively range through nth order theories by have nth order theory with a modified von Neumann Universe:

Why are you talking about semantics even though the OP is just intereted in syntax? It is completely irrelavant.

As I recommended you, you need to study formal logic before you talk as if you know it.