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<p>In set theory without the Axiom of Choice (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\textsf{AC}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">AC</mi> </math></EquationSource> </InlineEquation>), we <i>answer (in the negative) two open questions</i> from Lutz “Conway and Doyle Can Divide by Three, But I Can’t” and also study the deductive relationship between the statements “Every set divisible by <i>n</i> is strongly divisible by <i>n</i>” and “Every infinite set divisible by <i>n</i> has an infinite subset which is strongly divisible by <i>n</i>”, where <i>n</i> is any natural number <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we investigate the interrelations of the above two statements with several weak choice principles, providing both positive and independence results in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\textsf{ZF}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">ZF</mi> </math></EquationSource> </InlineEquation> (Zermelo–Fraenkel set theory without <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\textsf{AC}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">AC</mi> </math></EquationSource> </InlineEquation>) and in <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\textsf{ZFA}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">ZFA</mi> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\textsf{ZF}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">ZF</mi> </math></EquationSource> </InlineEquation> with atoms).</p>

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Divisibility vs. Strong Divisibility in Set Theory Without \(\textsf{AC}\)

  • 期刊论文

  • 发表日期：

    2025年11月19日

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  • Eleftherios Tachtsis

摘要

In set theory without the Axiom of Choice ( \(\textsf{AC}\) AC ), we answer (in the negative) two open questions from Lutz “Conway and Doyle Can Divide by Three, But I Can’t” and also study the deductive relationship between the statements “Every set divisible by n is strongly divisible by n” and “Every infinite set divisible by n has an infinite subset which is strongly divisible by n”, where n is any natural number \(\ge 2\) ≥ 2 . Furthermore, we investigate the interrelations of the above two statements with several weak choice principles, providing both positive and independence results in \(\textsf{ZF}\) ZF (Zermelo–Fraenkel set theory without \(\textsf{AC}\) AC ) and in \(\textsf{ZFA}\) ZFA ( \(\textsf{ZF}\) ZF with atoms).