[FOM] CH and mathematics

Colin McLarty colin.mclarty at case.edu
Thu Feb 7 20:11:59 EST 2008

I got into this thread when someone had asked a good question:  Why did
set theorists so readily accept the Well-founding axiom V=WF, rather
than regard it as a bad limitation on the universe of sets the way they
regard V=L?  Someone quoted Steele saying that V=WF does not rule out
any interesting structures.  And I made that more precise, and quite
general:  Adding V=WF does not rule out any structure at all up to
isomorphism , where "structure" means anything defined in Bourbaki's
theory of higher order structures, or even more general theories than that. 

Of course we must specify what axioms it is being added *to*.  Thomas
Forster is surely right that my claim

> Then, second, with the axiom of choice every set is isomorphic to a
> well-founded set.

requires that the base theory include replacement, as against my saying
it could be Zermelo set theory.  Okay, set theorists very largely do
accept replacement so I am happy to add it.

But he goes off point when he objects to saying the category of WF sets
and functions is equivalent to the category of all sets and functions in
V.  He wrote:

> 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.

The original question assumed Zermelo set theory.  This precludes
universal-set theories.

My claim is both more and less than the parallel point that Thomas offers:

> 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.

For the purpose of deciding what structures exist (up to isomorphism),
and given that the "mathematical community" does accept Zermelo set
theory, and choice, and replacement, we can say much more.  Not only do
V=WF and V=FH capture the same mathematics.  

Leaving them both out *also* captures the same mathematics!  

Well-founding and anti-well-founding alike make absolutely no difference
to what structures exist (up to isomorphism) once you accept Zermelo set
theory plus choice and replacement.

But I have no idea whether the preference for V=WF was an historical
accident.  I just know it was not about determining what structures
exist (up to isomorphism). 

Timothy Chow raised a different concern, saying my argument:

> makes sense if one regards the only
> "purpose" of set theory as providing a foundation
> for mathematics, in the sense that sets are there
> only to provide raw material for building arbitrary
> mathematical structures.

He suggests two other purposes: one is to find "which axioms are
*true*."  That is a huge question and I offer no answer here.  The other
is more specific.  He says:

> if one wants to apply set theory to contexts
> (other than f.o.m.) in which it is natural to
> consider non-well-founded sets, then V = WF is
> no longer "costless."

I wonder what would be an example.  Barwise and Etchemendy offer
non-well founded sets for the semantics of self-referential sentences,
but I have already published a paper showing how an
isomorphism-invariant form of their very semantics is simpler than the
form they give -- and easier to generalize to allow variant versions you
might like to study.  You can alter the semantics in various ways
without having to change your set theoretic foundation!  My article is
"Anti-foundation and self-reference" J. of Phil. Logic v. 22, 1993, 19--28.

The key point is that ZF works just fine with non-well-founded
relations, as long as you do not require one of those relations to be the
membership relation (or an iterate of the membership relation) on some
set.  And a requirement like that is irrelevant to every application I
have seen.

best, Colin

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