[FOM] throwing darts at natural numbers (rejoinder to Arnon Avron's reply)
joeshipman@aol.com
joeshipman at aol.com
Thu Aug 6 14:47:41 EDT 2009
-----Original Message-----
From: Timothy Y. Chow <tchow at alum.mit.edu>
****
On Thu, 6 Aug 2009, joeshipman at aol.com wrote:
> He doesn't need to understand Freiling's proof, he just needs to know
the
> statement of Freiling's result, to be able to say "in that case CH
sounds
> pretty dubious".
But he could also say, "in that case AC sounds pretty dubious."
****
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.
****
> The foundational significance of Freiling's argument is that it is
the
> most counterintuitive consequence of CH yet discovered.
I would call it a consequence of AC + CH rather than a consequence of
CH.
****
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.
****
> 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.)
-- JS
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