Scholarlink

<p>This paper investigates the Hausdorff dimension properties of chains and antichains in Turing degrees and hyperarithmetic degrees. Our main contributions are threefold: First, for antichains in hyperarithmetic degrees, we prove that every maximal antichain necessarily attains Hausdorff dimension 1. Second, regarding chains in Turing degrees, we establish the existence of a maximal chain with Hausdorff dimension 0. Furthermore, under the assumption that <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\omega _1=(\omega _1)^L\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ω</mi> <mn>1</mn> </msub> <mo>=</mo> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mi>L</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, we demonstrate the existence of such maximal chains with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\Pi ^1_1\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="normal">Π</mi> <mn>1</mn> <mn>1</mn> </msubsup> </math></EquationSource> </InlineEquation> complexity. Third, we extend our investigation to maximal antichains of Turing degrees by analyzing both the packing dimension and effective Hausdorff dimension.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

On the hausdorff dimension of maximal chains and antichains of turing and hyperarithmetic degrees

  • 期刊论文

  • 发表日期：

    2025年11月9日

查看全文
  • Sirun Song,
  • Liang Yu

摘要

This paper investigates the Hausdorff dimension properties of chains and antichains in Turing degrees and hyperarithmetic degrees. Our main contributions are threefold: First, for antichains in hyperarithmetic degrees, we prove that every maximal antichain necessarily attains Hausdorff dimension 1. Second, regarding chains in Turing degrees, we establish the existence of a maximal chain with Hausdorff dimension 0. Furthermore, under the assumption that \(\omega _1=(\omega _1)^L\) ω 1 = ( ω 1 ) L , we demonstrate the existence of such maximal chains with \(\Pi ^1_1\) Π 1 1 complexity. Third, we extend our investigation to maximal antichains of Turing degrees by analyzing both the packing dimension and effective Hausdorff dimension.