On the deductive strength of the Erdős–Dushnik–Miller theorem and two order-theoretic principles

Eleftherios Tachtsis
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Abstract

We provide answers to open questions from Banerjee and Gopaulsingh (Bull Pol Acad Sci Math 71: 1–21, 2023) about the relationship between the Erdős–Dushnik–Miller theorem (\(\textsf{EDM}\)) and certain weaker forms of the Axiom of Choice (\(\textsf{AC}\)), and we properly strengthen some results from Banerjee and Gopaulsingh (2023). We also settle a part of an open question of Lajos Soukup (stated in Banerjee and Gopaulsingh (2023) [Question 6.1]) about the relationship between the following two order-theoretic principles, which [as shown in Banerjee and Gopaulsingh (2023)] are weaker than \(\textsf{EDM}\): (a) “Every partially ordered set such that all of its antichains are finite and all of its chains are countable is countable” (this is known as Kurepa’s theorem), and (b) “Every partially ordered set such that all of its antichains are countable and all of its chains are finite is countable”. In particular, we prove that (b) does not imply (a) in \(\textsf{ZF}\) (i.e., Zermelo–Fraenkel set theory without \(\textsf{AC}\)). Moreover, with respect to (b), we answer an open question from Banerjee and Gopaulsingh (2023) about its relationship with the following weak choice form: “Every set is either well orderable or has an amorphous subset”; in particular, we show that (b) follows from, but does not imply, the latter weak choice principle in \(\textsf{ZFA}\) (i.e., Zermelo–Fraenkel set theory with atoms).

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关于厄尔多斯-杜什尼克-米勒定理的演绎强度和两个秩序理论原则
我们回答了 Banerjee 和 Gopaulsingh (Bull Pol Acad Sci Math 71: 1-21, 2023) 提出的关于厄尔多斯-杜什尼克-米勒定理 (\(textsf{EDM}\)) 和选择公理 (\(textsf{AC}\)) 的某些较弱形式之间的关系的公开问题,并适当加强了 Banerjee 和 Gopaulsingh (2023) 的一些结果。我们还解决了拉约什-苏库普(Lajos Soukup)提出的一个未决问题的一部分(在班纳吉和戈帕辛格(2023)[问题6.1]中提出),即下面两个有序理论原则之间的关系,这两个原则[如班纳吉和戈帕辛格(2023)所示]比\(\textsf{EDM}\)弱:(a) "每一个部分有序集合,如果它的所有反链都是有限的,而且它的所有链都是可数的,那么它就是可数的"(这被称为库雷帕定理),以及 (b) "每一个部分有序集合,如果它的所有反链都是可数的,而且它的所有链都是有限的,那么它就是可数的"。特别是,我们证明了(b)在(\textsf{ZF}\)中并不意味着(a)(即没有(\textsf{AC}\)的Zermelo-Fraenkel集合论)。此外,关于(b),我们回答了班纳吉和戈珀辛格(2023)提出的一个开放问题,即它与下面弱选择形式的关系:"每个集合要么是有序的,要么有一个无定形子集";特别是,我们证明了(b)是从(\textsf{ZFA}\)(即有原子的泽梅洛-弗伦克尔集合论)中的后一个弱选择原则得出的,但并不意味着后一个弱选择原则。
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