Ideal Analytic Sets

IF 0.4 4区数学 Q4 LOGIC

Mathematical Logic Quarterly Pub Date : 2026-02-27 DOI:10.1002/malq.70012

Łukasz Mazurkiewicz, Szymon Żeberski

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引用次数: 0

Abstract

The aim of this study is to give natural examples of $Σ_{1}^{1}$ -complete and $Π_{1}^{1}$ -complete sets.

In the first part, we consider ideals on $ω$ . We use a unified approach introduced in [4] to create reductions of the collection of ill-founded trees to the ideals, proving $Σ_{1}^{1}$ -completeness of the ideals.

In the second part, we show the connection between this topic, families of trees and coding of $σ$ -ideals of Polish spaces. In particular, we use the unified approach to prove that sets of codes for closed Ramsey-null sets, for closed $σ$ -compact sets and for closed not strongly dominating sets are $Π_{1}^{1}$ -complete.

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理想解析集

本研究的目的是给出Σ 11 $\operatorname{\mathbf {\Sigma }_1^1}$ -完备集和Π 11 $\operatorname{\mathbf {\Pi }_1^1}$ -完备集的自然例子。在第一部分中，我们考虑ω $\omega$上的理想。我们使用[4]中引入的统一方法来创建对理想的不正确树集合的约简，证明了Σ 1 $\operatorname{\mathbf {\Sigma }_1^1}$ -理想的完备性。在第二部分中，我们展示了这个主题、树族和波兰空间的σ $\sigma$ -理想编码之间的联系。特别地，我们用统一的方法证明了闭Ramsey-null集、闭σ $\sigma$ -紧集和闭非强支配集的码集是Π 11 $\operatorname{\mathbf {\Pi }_1^1}$ -完备的。

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

Mathematical Logic Quarterly 数学-数学

CiteScore

0.60

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Mathematical Logic Quarterly publishes original contributions on mathematical logic and foundations of mathematics and related areas, such as general logic, model theory, recursion theory, set theory, proof theory and constructive mathematics, algebraic logic, nonstandard models, and logical aspects of theoretical computer science.