[FOM] 2 questions

[William Everett Piper]

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