[FOM] CH and mathematics

Colin McLarty colin.mclarty at case.edu
Sat Feb 2 08:58:42 EST 2008

When Timothy Y. Chow asked why the well founding axiom V =WF is so readily
accepted, James Hirschorn <James.Hirschorn at univie.ac.at> replied

> The answer seems to be that V = WF does not exclude anything that
> most mathematicians care about.
> This agrees with [Steele's] answer, that "we know of no interesting
> structure outside WF".

But much more is true:  The well founding axiom does not exclude any
structure at all, up to isomorphism.  No isomorphism type of structure
in V lies outside WF.  Insofar as structures are only interesting up to
isomorphism, there are provably no interesting structures in V outside
of WF.

Compare V=L.  Every set in V is isomorphic to one in L, since every set
is well-orderable and all ordinals are in L.  But not every structure
existing in V is isomorphic to one in L.  

The most familiar example is to assume also a measurable cardinal k in V
and let F be a k-complete non-principle ultrafilter on k.  Of course k
is also in L since it is an ordinal.  Further, F as a set is isomorphic
to some set in L since it is isomorphic to some ordinal.  Even P(k) as
defined in V is isomorphic in V to a set in L although that set may be
strictly larger than (thus, not isomorphic to) the powerset of k as
defined in L.  But the inclusion function F>->P(k) which makes F an
ultrafilter on k does not exist in L.  Not even up to isomorphism. 

No function f:S-->T exists in L such that S is isomorphic in V to k, and
T isomorphic in V to the powerset P(k) as defined in V, and when you
take those isomorphisms back to V they carry F to f.

These facts on V=WF and V=L follow trivially from even more trivial
facts: Provably in ZF, every subset of a well-founded set is
well-founded.  But ZF does not prove not every subset of a constructible
set is constructible.

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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