Generic existence of interval P-points

IF 0.3 4区 数学 Q1 Arts and Humanities
Jialiang He, Renling Jin, Shuguo Zhang
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引用次数: 0

Abstract

A P-point ultrafilter over \(\omega \) is called an interval P-point if for every function from \(\omega \) to \(\omega \) 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 \(\textsf{CH}\) or \(\textsf{MA}\). (2) We identify a cardinal invariant \(\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})\) such that every filter base of size less than continuum can be extended to an interval P-point if and only if \(\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})={\mathfrak {c}}\). (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 \({\mathfrak {d}}={\mathfrak {c}}\) or \(\textbf{cov}({\mathcal {B}})={\mathfrak {c}}\).

区间p点的一般存在性
如果对于从\(\omega \)到\(\omega \)的每个函数,在这个超过滤器中存在一个集合A,使得函数对A的限制要么是一个常数函数,要么是一个区间到1的函数,那么\(\omega \)上的p点超过滤器就被称为区间p点。本文证明了以下结果。(1)在\(\textsf{CH}\)或\(\textsf{MA}\)条件下,区间p点不是同构不变的。(2)我们确定了一个基数不变量\(\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})\),使得每个小于连续统的滤波器基都可以扩展到区间p点,当且仅当\(\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})={\mathfrak {c}}\)。(3)在\({\mathfrak {d}}={\mathfrak {c}}\)或\(\textbf{cov}({\mathcal {B}})={\mathfrak {c}}\)假设下,证明了非拟选择性和非弱拉姆齐的慢/快区间p点和慢/快区间p点的一般存在性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Archive for Mathematical Logic
Archive for Mathematical Logic MATHEMATICS-LOGIC
CiteScore
0.80
自引率
0.00%
发文量
45
审稿时长
6-12 weeks
期刊介绍: The journal publishes research papers and occasionally surveys or expositions on mathematical logic. Contributions are also welcomed from other related areas, such as theoretical computer science or philosophy, as long as the methods of mathematical logic play a significant role. The journal therefore addresses logicians and mathematicians, computer scientists, and philosophers who are interested in the applications of mathematical logic in their own field, as well as its interactions with other areas of research.
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