[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
> gives the right answer.
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
> right answer.
But the next sentence is "This may have to do with the fact that in most cases
the other direction of the presumed independence result would involve set
theory at a substantially more sophisticated level.", suggesting something
like "fear/avoidance of the unknown" as opposed to rational consideration.
James Hirschorn
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