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<p>In Paolini and Shelah (2024), we proved that the space of countable torsion-free abelian groups is Borel complete. In this paper, we show that our construction from Paolini and Shelah (2024) satisfies several additional properties of interest. We deduce from this that countable torsion-free abelian groups are faithfully Borel complete; in fact, more strongly, we can <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11425_2024_2401_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathfrak{L}_{{\omega}_1},_\omega\)</EquationSource> </InlineEquation>-interpret countable graphs in them. Secondly, we show that the relation of pure embeddability (i.e., elementary embeddability) among countable models of Th(ℤ<sup>(<i>ω</i>)</sup>) is a complete analytic quasi-order.</p>

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Torsion-free abelian groups are faithfully Borel complete and pure embeddability is a complete analytic quasi-order

  • 期刊论文

  • 发表日期：

    2025年8月4日

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  • Gianluca Paolini,
  • Saharon Shelah

摘要

In Paolini and Shelah (2024), we proved that the space of countable torsion-free abelian groups is Borel complete. In this paper, we show that our construction from Paolini and Shelah (2024) satisfies several additional properties of interest. We deduce from this that countable torsion-free abelian groups are faithfully Borel complete; in fact, more strongly, we can \(\mathfrak{L}_{{\omega}_1},_\omega\) -interpret countable graphs in them. Secondly, we show that the relation of pure embeddability (i.e., elementary embeddability) among countable models of Th(ℤ(ω)) is a complete analytic quasi-order.