[FOM] Re: WHY DO SET THEORISTS DISLIKE CHRIS FREILING'S EVIDENCE AGAINST THE CONTINUUM HYPOTHESIS?
Timothy Y. Chow
tchow at alum.mit.edu
Thu Sep 2 09:50:54 EDT 2004
John McCarthy <jmc at steam.stanford.edu> wrote:
>I don't understand why almost all set theorists ignore, i.e. don't
>refer to Chris Freiling's paper offering evidence for the negation of
>the continuum hypothesis. It seems to me that G[f6]del would have
>considered it the kind of intuitive axiom he wanted.
If you're literally asking the sociological question of why the community
of set theorists ignores the paper, or doesn't find it convincing, then I
don't know. Why do most set theorists reject V = L? Why does Woodin like
projective determinacy while others are unconvinced? Is there anything
more to it than fashion or historical accident? These are hard questions.
On the other hand, if you're actually asking for reasons that people give
for finding Freiling's argument unconvincing, then at least I can give my
own opinion, which is influenced by Keith Devlin's article on the subject
( http://www.maa.org/devlin/devlin_6_01.html ). Freiling's argument
depends on assuming that the concept of randomness/probability/measure
applies to certain "weird" sets associated with a well-ordering of the
reals. We've all been indoctrinated in school about how the axiom of
choice lets us construct non-measurable sets, so I don't see why we should
believe that the particular weird sets in Freiling's argument should be
measurable.
A less formal but maybe more striking way to put it is that since the
axiom of choice implies something as weird as the Banach-Tarski paradox,
what's to stop it from implying something as weird as the continuum
hypothesis?
Tim
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