On Sat, Mar 15, 2008 at 11:31 AM, Thomas Streicher <streicher@mathematik.tu-darmstadt.de> wrote:
It might be the case that any "syntactically definable" external family indexed by EE(1,X) arises from some internal family a : A -> X though I don't see how to prove it.
I thought this was exactly what Colin's version of the replacement axiom says.
The reason why I doubt that models of ETCS and bZ are equivalent is that when going from a model of ECTS to a model of bZ is that one has to restrict to the well-founded part. At least that's what I recollect from MacLane and Moerdijk's exposition in their book. But I am ready to believe that adding wellfoundedness axioms to ETCS can remedy this situation.
Osius uses a different construction than M&M: instead of explicitly building "membership trees" he uses objects equipped with a transitive well-founded relation to represent transitive sets, and subobjects of them to represent arbitrary sets. In general you do have to add a "transitive representation" axiom to ensure that every object of the topos can be reconstructed from the resulting model of bZ---but this is unnecessary if you also assume choice, since then every object can be well-ordered and thus admits a transitive well-founded relation. Mike