Re: internal versus external

["Michael Shulman" <shulman@uchicago.edu>]

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