[FOM] V = WF costs nothing

Thomas Forster T.Forster at dpmms.cam.ac.uk
Mon Feb 4 17:53:59 EST 2008


I'm being cryptic, sorry.  The point is that there might be facts about 
large collections that cannot be presented (``spun'') as facts about 
wellfounded sets.  Mightn't there?   Church gave us a consistent set 
theory which says there is a universal set.  Are all facts about the 
universal set secretly facts about wellfounded sets?  Or is Church's set 
theory a snare and a delusion?  That is, it's consistent but not true..?
Possibly, but how can one tell?

   Don't get me wrong: i'm not knocking the idea that everything can be 
encoded in Th(WF): it's not a crazy idea, far from it.  It's a very 
interesting idea. I think of it as the project that Hilbert *should* have 
had, the correct generalisation of the idea that mathematics is the study 
of finite objects (= objects of finite character).  The only problem is, 
nobody has ever produced any arguments for it.

> So V=WF changes nothing about ordinary mathematics, all of which is done
> up to isomorphism.   V=L does change things up to isomorphism.
> James H adds:
> > I would add that CH is in fact far more "productive" than ~CH. (But
> > many believe it tends to give the wrong answers.) Indeed, it keeps
> > cropping up from time to time in mainstream mathematics.
> Well, many ZF set theorists believe it gives wrong answers.  But
> Devlin's book THE AXIOM OF CONSTRUCTIBILITY (Springer 1977) shows how
> time after time algebraists, topologists, and analysts tend to think it
> gives the right answer.
> Compare Nik Weaver's current article:
> > http://arxiv.org/abs/math/0604198v1
> which James cites.  The article says "it appears that C*-algebraists
> generally tend to regard a problem as solved when it has been answered
> using CH."  I.e. they take the answer that follows from CH to be the
> right answer.
> best, Colin
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