# Mediate cardinals

This is a talk in the CUNY Logic Workshop on 2024 March 22.

In the late 1910s Bertrand Russell was occupied with two things: getting into political truble for his pacifism and trying to understand the foundations of mathematics. His students were hard at work with him on this second occuptation. One of those students was Dorothy Wrinch. In 1923 she gave a characterization of the axiom of choice in terms of a generalization of the notion of a Dedekind-finite infinite set. 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 the main result of her 1923 paper, that AC is equivalent to the non-existence of what she termed mediate cardinals. I will also talk about some new independence results. The two main results are: (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 conistently be any class.