wlawvere@buffalo.edu Thursday, March 13, 2008 9:20 pm Wrote, with much else, about an equivalent to the axiom scheme of replacement in ETCS, and his mimeographed notes on it now reprinted at TAC http://138.73.27.39/tac/reprints/articles/11/tr11abs.html
Using the definable classes of sets smaller than any given set, the postulate that there are arbitrarily large such classes closed under arbitrary definable operations phi is proposed. I am not aware of any further studies of that postulate.
It is right at the end, on p. 34 of the reprint. To me, it is really a very nice stream-lined version of a reflection principle and deserves more of a look. Up to now I have tended to translate it into a replacement scheme because they are more familiar to my audience. But it is much more elegant as it cuts straight down to a very elegant isomorphism-invariant principle. best, Colin