FOM: the Borel universe

Kanovei kanovei at
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). 
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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