FOM: re: the Borel universe

[Kanovei]

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