关于相对化柴廷的Ω的更多结果

IF 0.6 2区数学 Q2 LOGIC

Annals of Pure and Applied Logic Pub Date : 2025-04-04 DOI:10.1016/j.apal.2025.103586

Liang Yu

引用次数: 0

摘要

我们证明，假设 ZF，并限制于任何 ≤T 点集，柴廷的ΩU:x↦ΩUx=∑Ux(σ)↓2-|σ| 对于任何通用无前缀图灵机 U 都不是注入式的，并且ΩUx 在非常强的意义上不具有度不变性，这回答了描述集合论中最近的几个问题。此外，我们还证明了在 ZF+AD 下，映射 x 到 x-random 的每个函数 f 都必须在图灵度的上锥上是不可数到一的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Some more results on relativized Chaitin's Ω

We prove that, assuming ZF, and restricted to any

\leq_{T}

-pointed set, Chaitin's

Ω_{U} : x \mapsto Ω_{U}^{x} = \sum_{U^{x} (σ) ↓} 2^{- | σ |}

is not injective for any universal prefix-free Turing machine U, and that

Ω_{U}^{x}

fails to be degree invariant in a very strong sense, answering several recent questions in descriptive set theory. Moreover, we show that under

ZF + AD

, every function f mapping x to x-random must be uncountable-to-one over an upper cone of Turing degrees.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

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.