This is an invited talk at the 120 Years of Choice conference on 12 July 2024.


In 1923 Dorothy Wrinch, a former student of Bertrand Russell, gave a characterization of the axiom of choice in terms of a generalization of Dedekind-finite infinite sets. Unfortunately, her career turned toward mathematical biology and her logical work was forgotten by history. This talk is part of a project of revisiting Wrinch’s work from a modern perspective. I will present her main result, that AC is equivalent to the non-existence of what she termed mediate cardinals. I will also talk about some new independence results: (1) the smallest $\kappa$ for which a $\kappa$-mediate cardinal exists can consistently be any regular $\kappa$ and (2) the collection of regular $\kappa$ for which exact $\kappa$-mediate cardinals exist can consistently be any class.