In the 1966 La Jolla conference volume appeared Lawvere's paper "The category of categories as a foundation for mathematics". Is it known just how much mathematics can be done using Lawvere's axiom scheme? My confusion on this point arises partly from a remark by Gray in his paper "The categorical comprehension scheme" in which he asserted that Lawvere's axioms couldn't do all that was claimed for them. Of course all of this predates the recognition that an elementary topos (with nn object) can be treated as a "universe of sets". I know that there has been some work (e.g. Joyal and Moerdijk) on constructing models of ZF set theory within categories with special properties. Can such categories be constructed using Lawvere's axioms? I hope this question makes sense but my skills in set theory and logic are pretty limited. Carl Futia 11-Mar-2005 13:01:41 -0400,22182;000000000001-00000000