FOM: the historical and logical pedigree of the Borel universe
kanovei at wminf2.math.uni-wuppertal.de
Tue Dec 16 07:34:59 EST 1997
>From: Stephen G Simpson <simpson at math.psu.edu>
>Something I didn't mention earlier: ATR_0 is strong enough to
> prove some advanced theorems on Borel sets.
people could be curious how far the induction
along wf trees allows to get. How about:
1) Kondo uniformization
2) Theorem that a Borel set with countable sections
is a countable union of Borel curves. (Ant its
remarkable generalization by Louveau.)
3) Theorem that "if all uncountable CA contain perfect
subsets then all A_2 are L measurable".
Note that 1) is different from many of your examples
as it needs 2nd rather than 1st periodicity theorem,
2) is also different as it needs 3rd periodicity, while
3) is an example of statement which has been profed
"by forcing" but there is no evidence that it goes
beyond second order PA (unlike e.g. the Borel determinacy).
Is there perhaps a comprehensive (and available) reference
for the whole setup ?
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