The Axiom of Choice and the Partition Principle from Dialectica Categories

Abstract

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.