[FOM] 2 questions
William Everett Piper
William.Piper at Colorado.EDU
Mon Dec 8 18:17:17 EST 2008
Hi all,
I have two unrelated questions to pose to the group. The first is due to Detlefsen and simply to satisfy my curiosity. The second is more a series of questions based on a particular construction.
I. Classify those Sigma^0_n extensions of Q (Robinson's Arithmetic) which can represent themselves, i.e. their natural proof predicate satsifies the derivability conditions, and which are not themselves complete with respect to Sigma^0_n predicates.
II. Let $I$ denote the set of all immune subsets of $\omega$, i.e. the set of all infinite subsets of natural numbers that do not contain an infinite r.e. set. Let $U$ be a non-principal ultrafilter on $I$ and suppose the set $X$ intersects every member of $U$. Does $X$ contain an infinite r.e. set?
Is $X\in P(\omega) - I$ or is $X\in I$? Could $X\in U$?
Best,
Everett Piper
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