[FOM] CH and mathematics

Thomas Forster T.Forster at dpmms.cam.ac.uk
Wed Feb 6 03:40:03 EST 2008

On Tue, 5 Feb 2008, Colin McLarty wrote:

> Then, second, with the axiom of choice every set is isomorphic to a
> well-founded set.  (The analog of this second step of course *does* hold
> for constructible sets.)

I don't think this is true.  That is to say i don't think that Coret's 
axiom (every set is the same size as a wellfounded set) follows from 
Choice.   It presumably does if you have replacement, so perhaps this 
is a quibble.

> As Hirschorn says this is categorical thinking: Altogether (provably in
> Zermelo set theory plus choice) the category of WF sets and functions is
> *equivalent* in the technical categorical sense to the category of all
> sets and functions in V.  

I think this, too, is a bit swift.  I'm no categorist so i have to tread
carefully.  I do have it on good authority that this allegation is true if
V is a Forti-Honsell antifoundation universe.  My guess is that it's not
true if V is a Church-Universal-set-theory universe.  If it is, then
equivalence of categories sounds like a very relaxed and tolerant
equivalence relation indeed.

There is a good point to be made here, and i think it is the one that 
Colin is trying to make.  Marco Forti likes to make the point that it 
really is pure historical accident that the mathematical community plumped 
for V = WF rather than V = a Forti-Honsell antifoundation universe, and 
that in some sense these two ways of doing set theory capture the same 
mathematics. (Another way of putting this is that  V = WF, and F-H 
antifoundation are two extensions of ZF-with-Coret's axiom that are 
conservative for stratified formulae).  All of this is true and important, 
but altho' this is what i think Colin *meant*, it isn't what he *said*.

> Colin
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