FOM: the Borel universe (a positive posting)
case at eecis.udel.edu
Sun Dec 7 16:11:32 EST 1997
On Dec 7, 14:38, Stephen G Simpson wrote:
} Subject: Re: FOM: the Borel universe (a positive posting)
Date: Sun, 7 Dec 1997 14:38:29 -0500 (EST)
From: Stephen G Simpson <simpson at math.psu.edu>
To: John Case <case at eecis.udel.edu>
Subject: Re: FOM: the Borel universe (a positive posting)
John Case writes:
> Steve, this material is clearly neat, but what's wrong with well-orderings
> of the continuum? (-8 John
What's wrong with them is that they lead to a lot of pathology:
non-measurable sets, the Banach-Tarski paradox, etc. This seems to
have been on the minds of the "Cinq lettres" group.
}-- End of excerpt from Stephen G Simpson
I edited out some mail header in the above. I may have hit the wrong replay
button some time back and sent my ``what's wrong with message'' only to Steve.
Re non-measurable sets: It seems unlikely to me that beautiful attempts
such as measurability to extend the notion of lengths of lines, rectangles and
the like to possibly most general classes of sets would necessarily extend to
all the sets one can think of. I am, nonetheless, impressed by the Borel sets
being measurable and especially their providing game determinancy.
Banach-Tarski is a little tougher, but, again, there is no apriori reason to
expect concepts of volume to apply to any old sets one might dream up.
Vladimir Kanovei's examples may be neat too. I'm not close enough to
equivalence relations on countable models, etc. to have a knowlegeable opinion.
(-8 John Case
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