Jens Mammen (Professor Emeritus of psychology at Aarhus and Aalborg University) has developed a theory in psychology, which aims to provide a model for the interface between a human being (and mind), and the real world.
This theory is formalized in a very mathematical way: Indeed, it is described through a mathematical axiom system. Realizations ("models") of this axiom system consist of a non-empty set U (the universe of objects), as well as a perfect Hausdorff topology S on U, and a family C of subsets of U which must satisfy certain axioms in relation to S. The topology S is used to model broad categories that we sense in the world (e.g., all the stones on a beach) and the C is used to model the process of selecting an object in a category that we sense (e.g., a specific stone on the beach that we pick up). The most desirable kind of model of Mammen's theory is one in which every subset of U is the union of an open set in S and a set in C. Such a model is called "complete".
The harder mathematical aspects of Mammen's theory were first studied in detail by J. Hoffmann-Joergensen in the 1990s. Hoffmann-Joergensen used the Axiom of Choice (AC) to show that a complete model of Mammen's axiom system, in which the universe U is infinite, does exist. Hoffmann-Joergensen conjectured at the time that the existence of a complete model of Mammen's axioms would imply the Axiom of Choice.
I will discuss the set-theoretic aspects of complete Mammen models. First of all, the question of "how much" AC is needed to obtain a complete Mammen model; secondly, I will introduce some cardinal invariants related to complete Mammen models and establish elementary ZFC bounds for them, as well as some consistency results.
This is joint work with Jens Mammen.