Scholarlink

<p>We prove, in <Emphasis FontCategory="SansSerif">ZFC</Emphasis>, that certain <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varvec{\Sigma }^1_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mi mathvariant="bold">Σ</mi> </mrow> <mn>2</mn> <mn>1</mn> </msubsup> </math></EquationSource> </InlineEquation> functions cannot injectively embed <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\omega _1\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ω</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> into a Borel class of fixed countable rank. This had been proved under determinacy or large cardinals by Harrington and Hjorth for all <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\varvec{\Sigma }^1_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mi mathvariant="bold">Σ</mi> </mrow> <mn>2</mn> <mn>1</mn> </msubsup> </math></EquationSource> </InlineEquation> functions. Our contribution is to identify conditions under which the determinacy and large cardinal assumptions can be removed. These conditions are sufficient for a recent use of the non-existence of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\varvec{\Sigma }^1_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mi mathvariant="bold">Σ</mi> </mrow> <mn>2</mn> <mn>1</mn> </msubsup> </math></EquationSource> </InlineEquation> injections of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\omega _1\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ω</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> into Borel classes by Day and Marks.</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

A further application of Stern absoluteness

  • 期刊论文

  • 发表日期：

    2026年7月1日

查看全文
  • Itay Neeman

摘要

We prove, in ZFC, that certain \(\varvec{\Sigma }^1_2\) Σ 2 1 functions cannot injectively embed \(\omega _1\) ω 1 into a Borel class of fixed countable rank. This had been proved under determinacy or large cardinals by Harrington and Hjorth for all \(\varvec{\Sigma }^1_2\) Σ 2 1 functions. Our contribution is to identify conditions under which the determinacy and large cardinal assumptions can be removed. These conditions are sufficient for a recent use of the non-existence of \(\varvec{\Sigma }^1_2\) Σ 2 1 injections of \(\omega _1\) ω 1 into Borel classes by Day and Marks.