[FOM] throwing darts at natural numbers (rejoinder to Arnon Avron's reply)

[Timothy Y. Chow]

On Thu, 6 Aug 2009, joeshipman at aol.com wrote:
> I disagree. The "random mathematician" you posit doesn't need to know 
> anything about well-orderings to have an intuition that when iterated 
> integrals exist the order shouldn't matter, or the equivalent but even 
> more compelling intuition that sets of measure 0 on every horizontal 
> line should not have full measure on every vertical line, or the weaker 
> but yet more compelling intuition that sets with countably many points 
> on every horizontal line should not have co-countably many points on 
> every vertical line.

I meant to comment on this too.  Your point that only the conclusion of 
Freiling's argument needs to be grasped, and not the details of the proof, 
is well taken.  Nevertheless, I still don't believe that a random 
mathematician has the intuitions you attribute here.

For Fubini's theorem, I suspect that most people's intuition is that you 
need some technical conditions to rule out pathologies.

For the other two results you mention, if you posed the question, I 
suspect that most people's reaction would be, "I don't know...let me think 
for a minute...hmmm..." and then try to find a proof or a counterexample.
Again, they would probably feel that *nice* sets wouldn't behave that way, 
but that there might or might not be pathological sets that behave in some 
counterintuitive fashion.

Tim