[FOM] throwing darts at natural numbers (rejoinder to Arnon Avron's reply)
Timothy Y. Chow
tchow at alum.mit.edu
Thu Aug 6 13:29:22 EDT 2009
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
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