[FOM] V = WF costs nothing
Thomas Forster
T.Forster at dpmms.cam.ac.uk
Mon Feb 4 17:53:59 EST 2008
Colin:
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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