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Abstract <p> We prove that the full Comprehension schema <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3031_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbf{CA}\)</EquationSource> </InlineEquation> in second-order arithmetic <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3031_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbf{PA}_2\)</EquationSource> </InlineEquation> is not provable in the subtheory <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3031_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbf{PA}_2^\ast\)</EquationSource> </InlineEquation> with the parameter-free Comprehension even when adding the parameter-free Countable Choice <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3031_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbf{AC}_\omega^\ast\)</EquationSource> </InlineEquation> and the Comprehension <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3031_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbf{CA}(\mathbf\Sigma^1_2)\)</EquationSource> </InlineEquation> for all <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3031_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbf\Sigma^1_2\)</EquationSource> </InlineEquation> formulas with parameters. </p>

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Independence of the Comprehension Schema in Second-Order Arithmetic from the Parameter-Free Countable Choice

  • 期刊论文

  • 发表日期：

    2025年2月1日

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  • V. G. Kanovei,
  • V. A. Lyubetsky

摘要

Abstract

We prove that the full Comprehension schema \(\mathbf{CA}\) in second-order arithmetic \(\mathbf{PA}_2\) is not provable in the subtheory \(\mathbf{PA}_2^\ast\) with the parameter-free Comprehension even when adding the parameter-free Countable Choice \(\mathbf{AC}_\omega^\ast\) and the Comprehension \(\mathbf{CA}(\mathbf\Sigma^1_2)\) for all \(\mathbf\Sigma^1_2\) formulas with parameters.