FOM: re: the Borel universe
Kanovei
kanovei at wminf2.math.uni-wuppertal.de
Thu Dec 11 10:14:58 EST 1997
ref. Stephen G Simpson Sat, 6 Dec 1997 17:46:34 -0500 (EST)
>I now propose an elegant axiomatization of a significant part of the
>theory of the Borel universe. Let's temporarily call it TBU_0.
(This theory intends to deal with Borel sets only,
treating results on analytic sets, like Separation,
in some indirect way.)
Those working in DST hardly can abandon free access
to analytic and co-analytic sets. A more plausible
approach would be to use a language which does NOT
contain quantifiers over reals, but contains
quantifiers which simulate A-operation or more
general game operations
(Shilling and Vaught, Trans. AMS, 1983, 279, 411 - 428).
This naturally leads to a collection of sets of reals
(containing e.g. all analytic and co-analytic sets
and contained in 2nd projective level),
where different paradoxes related to measure
(in particular Banach - Tarcki) do not occur
and classical DST works.
(Apart of a few results like Borel determinacy.
There can also be technical problems in
the full elimination of projection in results like that
Sigma^1_2 sets are aleph_1-unions of Borel sets.)
A concrete axiomatization may be not so easy, of course.
Vladimir Kanovei
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