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Abstract <p>The paper investigates computability-theoretic aspects of some model-theoretic results. We study the Weihrauch degrees for problems of finding an elementary embedding between models that are elementarily equivalent. We prove that each of the following two problems is strongly Weihrauch equivalent to the problem <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\lim\)</EquationSource> <!--LobJMat2561383Bazhenov-m1--> </InlineEquation> (i.e., the limit map in Baire space): finding an elementary embedding from a prime model <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal{M}\)</EquationSource> <!--LobJMat2561383Bazhenov-m2--> </InlineEquation> into an arbitrary model <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathcal{N}\)</EquationSource> <!--LobJMat2561383Bazhenov-m3--> </InlineEquation>, and finding an elementary embedding from an arbitrary model <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathcal{M}\)</EquationSource> <!--LobJMat2561383Bazhenov-m4--> </InlineEquation> into a countable saturated model <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathcal{N}\)</EquationSource> <!--LobJMat2561383Bazhenov-m5--> </InlineEquation>.</p>

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Weihrauch Degrees of Elementary Embeddings for Prime and Saturated Models

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • N. A. Bazhenov,
  • M. Fiori-Carones,
  • M. I. Marchuk

摘要

Abstract

The paper investigates computability-theoretic aspects of some model-theoretic results. We study the Weihrauch degrees for problems of finding an elementary embedding between models that are elementarily equivalent. We prove that each of the following two problems is strongly Weihrauch equivalent to the problem \(\lim\) (i.e., the limit map in Baire space): finding an elementary embedding from a prime model \(\mathcal{M}\) into an arbitrary model \(\mathcal{N}\) , and finding an elementary embedding from an arbitrary model \(\mathcal{M}\) into a countable saturated model \(\mathcal{N}\) .