FOM: the Borel universe
Kanovei
kanovei at wminf2.math.uni-wuppertal.de
Sun Dec 7 12:11:03 EST 1997
There are objects in real mathematics which
are analytic but not Borel,
as a hot example, equivalence relations on
countable models are analytic but
usually non-Borel
(see Hjorth and Kechris, JSL, 1995, no 4).
But the number of equivalence classes of an
analytic equivalence relation on R can be
a counterexample to CH. Thus it hardly can
be expected that moderate and
mathematically reasonable restrictions of
the set universe will allow us to "solve"
CH this way.
As a comment to the tilt towards Borel sets
in Cinq Lettres,
the participants of the discussion perhaps
could not imagine the existence of interesting
individual non-Borel sets in R (like
analytic or co-analytic).
Indeed
the 1st example of a non-Borel
individual set was given by Lebesgue in 1905
(J. de math., 1905, 1),
analytic sets were introduced in 1917,
and during some time the wrong statement of
Lebesgue that Borel sets are closed under
projection hinted that Borel sets are better
than they are.
Vladimir Kanovei
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