# [FOM] CH and mathematics

James Hirschorn James.Hirschorn at univie.ac.at
Sun Feb 3 22:40:39 EST 2008

```On Saturday 02 February 2008 08:58, Colin McLarty wrote:
> 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.

This is the best explanation I have seen for the "harmlessness" of V=WF. I
assume that you mean "structure" in the formal sense (i.e. a triple (A,S,I),
where A is the domain, S the signature and I the interpretation), in the
theorem:

Theorem (ZF - Foundation + AC). Every structure is isomorphic to a structure
in WF.

In Kunen's book (Lemma 2.14, pp. 98) he mentions this, but only for groups and
topological spaces. He also points out a possible caveat: The above Theorem
is false without AC.

>
> 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
>
> ...
>
> 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.

To be precise, you presumably meant "ZF - Foundation" to avoid a tautology.

> 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.

Yes, and I think this is an example of the value of the category theoretic
viewpoint to foundations.

>
> 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

I look forward to checking whether there is any basis for their belief that CH
is giving the right answer, the next time have this book in my possesion.

> 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