[FOM] CH and mathematics32
James Hirschorn
James.Hirschorn at univie.ac.at
Thu Jan 31 22:31:20 EST 2008
> Arnon Avron wrote:
> > I am not arguing about that. The only point I wanted to make at the
> > above quoted paragraph is that it is intelectually dishonest to reject
> > one axiom (V=L) because it seems to be FALSE, but to argue for another
> > one (-CH) because there are reasons to view it as more productive than
> > its negation.
>
Timothy Y. Chow replied:
> This reminds me of another issue that has bothered me, which is the status
> of the axiom of regularity, which I'll write as "V = WF." How is it that
> V = WF has managed to get accepted so widely with so little fuss? It is
> not obviously "true" and it restricts the universe, so shouldn't the usual
> criticisms of V = L also apply to it?
This was discussed e.g. by Steel in some talk for which slides are available
("laguna.ps").
>
> The answer seems to be that V = WF does not exclude anything that most
> mathematicians care about.
This agrees with his answer, that "we know of no interesting structure
outside WF".
>
> In light of this observation, my feeling is that people who reject V = L
> reject it not because it is "false" (even if that's what they say), but
> because adopting it eliminates a lot of phenomena that they find very
> interesting. (I suppose they could, in principle, develop the theory of
> such phenomena *within* ZFC + V=L, but that would be cumbersome.)
This was also discussed, that ZFC + V=L lacks the expressive power to fully
develop the theory of e.g. measurable cardinals, although *some* classes of
consequences can be developed (the example in the slides is all Sigma-1-2
consequences of the existence of a measurable cardinal). Thus, up to my
understanding, it really does eliminate interesting phenomena.
>
> And getting back to ~CH, I think the argument from "productivity" is
> unlikely to carry the day by itself. Instead, what has to happen is for
> some ("productive") axiom to (1) imply ~CH and (2) exclude nothing of
> "interest" if it is adopted.
>
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. For example, there is
Kaplansky's conjecture on Banach algebras, which was originally answered by
an analyst (Dales) using CH who (if I remember correctly) thought that the
use of CH was just a weakness of his proof; but in fact turned out to be
independent of ZFC (proved by Solovay--Woodin, I think). It has also been
repeatedly used in operator theory. See Nik Weaver's article:
http://arxiv.org/abs/math/0604198v1
Martin Davis wrote:
> > Mathematicians (Weyl, Borel) famously did worry about these concepts
> > in the early years of the 20th century. But analysts have been
> > cheerfully working with them for many decades with no dificulties.
Arnon Avron replied:
> They have also been cheerfully working for many decades
> without caring a bit about FOM - and with no difficulties.
Not so. There are occasionally difficulties; some examples are mentioned
above.
James Hirschorn
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