[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 
> statement of Freiling's result, to be able to say "in that case CH 
> 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 
> most counterintuitive consequence of CH yet discovered.

I would call it a consequence of AC + CH rather than a consequence of 

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 

> 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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