Fundamental sequences based on localization

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic Pub Date : 2025-09-24 DOI:10.1016/j.apal.2025.103658

Gunnar Wilken

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引用次数: 0

Abstract

Building on Buchholz' assignment for ordinals below Bachmann-Howard ordinal, see [2], we introduce systems of fundamental sequences for two kinds of relativized ϑ-function-based notation systems of strength

Π_{1}^{1} {-CA}_{0}

and prove Bachmann property for these systems, which is essential for monotonicity properties of subrecursive hierarchies defined on the basis of fundamental sequences. The central notion of our construction is the notion of localization, which was introduced in [12].

The first kind of stepwise defined ϑ-functions over ordinal addition as basic function fits the framework of the ordinal arithmetical toolkit developed in [12], whereas the second kind of ϑ-functions is defined simultaneously and will allow for further generalization to larger proof-theoretic ordinals, see [10].

The systems of fundamental sequences given here enable the investigation of fundamental sequences and independence phenomena also in the context of patterns of resemblance, an approach to ordinal notations that is both semantic and combinatorial and was first introduced by Carlson in [4] and further analyzed in [11], [13], [14], [5].

Our exposition is put into the context of the abstract approach to fundamental sequences developed by Buchholz, Cichon, and Weiermann in [3]. The results of this paper will be applied to the theory of Goodstein sequences, extending results of [7].

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基于定位的基本序列

在Buchholz对低于Bachmann- howard序数的赋值（见[2]）的基础上，我们引入了两种强度为Π11-CA0的相对ϑ-function-based符号系统的基本序列系统，并证明了这些系统的Bachmann性质，这对于基于基本序列定义的子递归层次的单调性是必不可少的。我们构建的中心概念是本地化的概念，这是在b[12]中引入的。第一类在序数加法上逐步定义的ϑ-functions作为基本函数符合[12]中开发的序数算术工具包的框架，而第二类ϑ-functions是同时定义的，并且将允许进一步推广到更大的证明论序数，参见[10]。这里给出的基本序列系统使得在相似模式的背景下研究基本序列和独立现象成为可能，相似模式是一种语义和组合的有序符号方法，由Carlson在[4]中首次引入，并在[11]，[13],[14]，[5]中进一步分析。我们的阐述被置于由Buchholz， Cichon和Weiermann在1986年开发的基本序列的抽象方法的背景下。本文的结果将应用于Goodstein序列理论，推广了[7]的结果。

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

Annals of Pure and Applied Logic 数学-数学

CiteScore

1.40

自引率

12.50%

发文量

审稿时长

200 days

期刊介绍： The journal Annals of Pure and Applied Logic publishes high quality papers in all areas of mathematical logic as well as applications of logic in mathematics, in theoretical computer science and in other related disciplines. All submissions to the journal should be mathematically correct, well written (preferably in English)and contain relevant new results that are of significant interest to a substantial number of logicians. The journal also considers submissions that are somewhat too long to be published by other journals while being too short to form a separate memoir provided that they are of particular outstanding quality and broad interest. In addition, Annals of Pure and Applied Logic occasionally publishes special issues of selected papers from well-chosen conferences in pure and applied logic.