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Kleene degrees of ultrafilters. (English) Zbl 0573.03020

Recursion theory week, Proc. Conf., Oberwolfach/Ger. 1984, Lect. Notes Math. 1141, 29-48 (1985).
[For the entire collection see Zbl 0566.00001.]
When is a non-principal ultrafilter \({\mathcal V}\) on \(\omega\) Kleene reducible to (i.e., recursive in a real and) another such ultrafilter \({\mathcal U}?\) A sufficient condition is that \({\mathcal V}\) belongs to the smallest class of ultrafilters that contains \({\mathcal U}\) and is closed under taking images (under functions \(\omega\) \(\to \omega)\) and under taking limits with respect to ultrafilters in the class. It is conjectured that this condition is also necessary. The class described in the condition is studied in some detail; for example, it is linearly ordered under the Rudin-Keisler ordering if \({\mathcal U}\) is selective. The following special case of the conjecture is proved: If a selective ultrafilter \({\mathcal V}\) is Kleene reducible to a selective ultrafilter \({\mathcal U}\), then \({\mathcal U}\) and \({\mathcal V}\) are isomorphic.

MSC:

03E05 Other combinatorial set theory
03D65 Higher-type and set recursion theory

Citations:

Zbl 0566.00001