[FOM] throwing darts at natural numbers (rejoinder to Arnon Avron's reply)
Timothy Y. Chow
tchow at alum.mit.edu
Thu Aug 6 20:56:12 EDT 2009
On Thu, 6 Aug 2009, joeshipman at aol.com wrote:
> I consider the battle about AC to be over among "ordinary
> mathematicians", who are the ones we are talking about here. The
> consequences of rejecting AC have been generally evaluated as even
> weirder and more counterintuitive than the consequences of accepting AC,
> and intuitions have been revised accordingly.
[...]
> So you do not accept AC in the same way you accept the other ZF axioms?
> That's fine, but it's not the position of the mathematician in the
> street.
The battle is over in the sense that people understand AC fairly well, and
won't complain if you invoke it when you need it. This doesn't mean that
even its most extreme consequences are accepted as normal. Banach-Tarski
is still regarded as pathological.
Let's assume for the sake of argument that the examples you get by
assuming CH along with AC are "even weirder pathologies" than
Banach-Tarski. I don't think the mathematician in the street will
respond, "Gee, since I accept AC as gospel, I am forced to blame these
pathologies entirely on CH!"
As for whether I accept AC "in the same way I accept the other ZF axioms,"
what are you assuming about the way I accept the other ZF axioms? That I
accept them as definitely true? I think my attitude is more that I accept
them as needed. Most of the time I don't need Foundation or Replacement.
In this sense, AC is no different. More to the point, I would say that
the mathematician in the street does indeed treat AC differently from the
other axioms of ZF. For starters, said mathematician is unlikely to be
able even to list the axioms of ZF, but he or she will know AC explicitly,
precisely because it is known to have some strange consequences.
> ****
> > What is the most counterintuitive consequence of not-CH that has been
> > discovered so far?
>
> This is not exactly a "fair fight." Not-CH by itself is such a weak
> assertion that we don't expect it to have particularly interesting
> consequences on its own.
> ****
>
> OK, then take a stronger statement such as the "strong Fubini theorem in n
> dimensions" that I analyzed in my thesis:
>
> If f is a non-negative function on R^n such that the iterated Lebesgue
> integrals exist for 2 different orders of integration, then those iterated
> integrals have the same value.
>
> That is a proposed new axiom. It contradicts CH; but does it contradict
> intuition in any way? (It is an acceptable response to give an intuitive
> reason to believe CH is true.)
I haven't thought about it too hard, but it doesn't seem immediately
counterintuitive. Anyway, the conversation is now drifting away from the
question of whether Dunion has been misinterpreted, and into the more
general question of whether one can "settle" CH, which I'm not especially
interested in right now.
Tim
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