FOM: Borel sets
Kanovei
kanovei at wminf2.math.uni-wuppertal.de
Sun Dec 14 06:13:33 EST 1997
> I have proposed to restrict attention to Borel sets
Steve
there are proofs in descriptive set theory (eg one
of known proofs of the Suslin theorem) which go on
as follows.
1) We assume the negation and work towards contradiction.
2) At some step, we get a certain tree consisting of finite
sequences of integers.
3) We prove, using the assumption at step 1), that the tree
has an infinite branch.
4) The branch leads to contradiction which proves the result.
5) An inspection of step 3) shows, however, that to define
a branch we usually need Comprehension for formulas
which are combinations of Sigma11 and Pi11.
Conclusion: you are going to admit only Borel definitions
for *sets of reals* but you will most likely have to admit
essentially non-Borel definitions for *reals* themselves,
which does not look entirely consistent.
(Perhaps my observation is silly if there exist proofs
of the Suslin theorem and other similar results in the
"predicative" 2nd order PA. I don't know such.)
Vladimir Kanovei
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