The method of morphisms is a well-known application of Dialectica categories to set theory (more precisely, to the theory of cardinal invariants of the continuum). In a previous work, Valeria de Paiva and the author have asked how much of the Axiom of Choice is needed in order to carry out the referred applications of such method. In this paper, we show that, when considered in their full generality, those applications of Dialectica categories give rise to equivalents (within |$\textbf{ZF}$|⁠) of either the Axiom of Choice (⁠|$\textbf{AC}$|⁠) or Partition Principle (⁠|$\textbf{PP}$|⁠)—which is a consequence of |$\textbf{AC}$| whose precise status of its relationship with|$\textbf{AC}$| itself is an open problem for more than a hundred years.

中文翻译：

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选择公理和辩证法范畴的划分原则

态射的方法是Dialectica类别在设置理论（更确切地说，对于连续统的基本不变式理论）方面的众所周知的应用。在先前的工作中，瓦莱里亚·德·帕瓦（Valeria de Paiva）和作者曾问过需要多少选择公理才能进行这种方法的引用应用。在本文中，我们显示出，当以它们的普遍性考虑时，方言类别的那些应用会产生选择公理（⁠| $ \ textbf {的| $ \ textbf {ZF} $ |⁠）的等价物。 AC} $ |⁠）或分区原理（⁠| $ \ textbf {PP} $ |⁠）—这是| $ \ textbf {AC} $ |的结果。与| $ \ textbf {AC} $ |的关系的确切状态 本身是一个开放的问题，已有一百多年的历史了。