Bounded Zermelo set theory, and many variants named for MacLane in some way, are used in equiconsistency proofs for Simple Theory of Types plus infinity, and for the Elementary Theory of the Category of Sets starting in 1969 (see [Finite order arithmetic and ETCS](https://mathoverflow.net/questions/115207/finite-order-arithmetic-and-etcs)). But did anyone formulate it before that? 


Of course Bounded Zermelo could easily have been stated before that, and the analogy with Primitive Recursive Arithmetic could have suggested it. But was it formulated earlier in fact?



When was Bounded Zermelo set theory first formulated?
The Princeton thesis of John Kemeny, written in 1949, was devoted to the relation between Zermelo set theory and type theory. For example, early in the thesis, there is a proof of the consistency of the simple theory of types relative to the consistency of a small fragment (nowadays known as KF) of Bounded Zermelo set theory. The comprehension scheme in KF is limited to *stratifiable* instances of bounded comprehension. KF was explicitly introduced and studied by Kaye and Forster in their 1991 paper in JSL entitled "End-extensions Preserving Power Set". 


For more detail, see subsection 8.0 of Mathias' *Strength of Mac Lane Set Theory* (APAL, 2001); indeed Mathias further discusses Kemeny's thesis in detail towards the end of section 8 of his paper.


Based on Mathias' account, it is clear that, conceptually, Kemeny "knew about" bounded Zermelo set theory, but it is not clear to me whether he explicitly formulated any fragments of Zermelo set theory.


