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

joeshipman@aol.com joeshipman at aol.com
Thu Aug 6 01:00:55 EDT 2009

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.

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".

The foundational significance of Freiling's argument is that it is the 
most counterintuitive consequence of CH yet discovered. What is the 
most counterintuitive consequence of not-CH that has been discovered so 

-- JS

-----Original Message-----
From: Timothy Y. Chow <tchow at alum.mit.edu>
Dunion is suggesting that the typical mathematician who does not 
have a strong interest in f.o.m. still finds Freiling's argument
persuasive.  But in my experience, if you pick a random mathematician 
is not already interested in f.o.m., there's at least a 50% chance that
you'll have to remind them of the definition of a well-ordering of the
reals and of its relationship to the axiom of choice.  Thus I am 
of any claim that said mathematician has well-developed intuitions 
Freiling's argument.

Remember, after all, that Dunion's stated goal was not to take sides in 
well-known debate within the f.o.m. community, but to build a bridge 
a perceived gulf between the f.o.m. community and the rest of the
mathematical community.

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