[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
far?
-- JS
-----Original Message-----
From: Timothy Y. Chow <tchow at alum.mit.edu>
Dunion is suggesting that the typical mathematician who does not
already
have a strong interest in f.o.m. still finds Freiling's argument
persuasive. But in my experience, if you pick a random mathematician
who
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
doubtful
of any claim that said mathematician has well-developed intuitions
about
Freiling's argument.
Remember, after all, that Dunion's stated goal was not to take sides in
a
well-known debate within the f.o.m. community, but to build a bridge
over
a perceived gulf between the f.o.m. community and the rest of the
mathematical community.
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