Generic existence of interval P-points

A P-point ultrafilter over ω is called an interval P-point if for every function from ω to ω there exists a set A in this ultrafilter such that the restriction of the function to A is either a constant function or an interval-to-one function. In this paper we prove the following results. (1) Interva...

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Bibliographic Details
Published inArchive for mathematical logic Vol. 62; no. 5-6; pp. 619 - 640
Main Authors He, Jialiang, Jin, Renling, Zhang, Shuguo
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2023
Springer Nature B.V
Subjects
Algebra
Invariants
Isomorphism
Mathematical Logic and Foundations
Mathematics
Mathematics and Statistics
Interval P-point
Weakly Ramsey ultrafilter
P-point
Generic existence
03E05
Quasi-selective ultrafilter
03E65
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Summary:A P-point ultrafilter over ω is called an interval P-point if for every function from ω to ω there exists a set A in this ultrafilter such that the restriction of the function to A is either a constant function or an interval-to-one function. In this paper we prove the following results. (1) Interval P-points are not isomorphism invariant under CH or MA . (2) We identify a cardinal invariant non ∗ ∗ ( I int ) such that every filter base of size less than continuum can be extended to an interval P-point if and only if non ∗ ∗ ( I int ) = c . (3) We prove the generic existence of slow/rapid non-interval P-points and slow/rapid interval P-points which are neither quasi-selective nor weakly Ramsey under the assumption d = c or cov ( B ) = c .
ISSN:0933-5846
1432-0665
DOI:10.1007/s00153-022-00853-3