Re: question to Colin about uniqueness in his Replacement axiom

[Colin McLarty <colin.mclarty@case.edu>]

Thomas Streicher <streicher@mathematik.tu-darmstadt.de>
Tuesday, March 18, 2008 6:09 pm

Wrote:


> In your Replacement axiom (p.48 of your "Philosophia" article) you
> psotultathe existence of a map f : S -> A such that S_x \cong x^*f
> for all x : 1->X.
> Can you prove that this f is unique up to isomorphism, i.e. that
> wellpointedness for maps entails wellpointedness for families?


Sure.  It takes the axiom of choice of course, since without choice the
result may be false (even two countably infinite families of countably
infinite sets need not be isomorphic).

It is the obvious argument by Zorn's lemma, which follows from choice:

Given two families S-->A and S'-->A with corresponding fibers
isomorphic, consider the set of all pairs <U,i> with U a subset of A,
and i an isomorphism over U from the restriction of S to the restriction
of S'.  By Zorn at least one of these is maximal (for the obvious
ordering by inclusion) so call it <U,i>.  Since well-pointedness implies
Boolean, U has a complement in A--which must be empty or else we could
extend the isomorphism.

best, Colin