[FOM] throwing darts at natural numbers (rejoinder to Arnon Avron's reply)
Timothy Y. Chow
tchow at alum.mit.edu
Wed Aug 5 14:05:44 EDT 2009
Joe Shipman wrote:
>Dunion (and Freiling) have been misintrepreted. They are not claiming
>to have an argument formalizable in ZFC; they are merely claiming that
>mathematicians have overreacted to the results of Banach-Tarski, Godel,
>and Cohen by throwing out too much of their intuition about assigning
>measures to subsets of R^n. If there are going to be any generally
>accepted additions to the standard ZFC axioms that settle CH (and if
>you don't believe such additions are desirable then I'd like to know
>why), they are going to have to have some kind of intuitive appeal, and
>Freiling and Dunion are doing exactly what one would expect needs to be
>done to make progress on this.
I don't believe that Dunion has been misinterpreted. The program you
sketch here falls squarely within the f.o.m. realm. If you think it's an
interesting question to investigate plausible extensions of ZFC that
settle CH, then you're already a dyed-in-the-wool f.o.m.er.
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.
Tim
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