Set theory and a model of the mind in psychology

TL;DR

This paper explores the mathematical structure of a psychological model called Mammen spaces, demonstrating their properties under various set-theoretic assumptions and refuting a conjecture linking their completeness to the Axiom of Choice.

Contribution

It proves the existence of complete Mammen spaces in models where the Axiom of Choice fails and introduces new cardinal invariants related to these spaces.

Findings

Complete Mammen spaces exist in the Cohen model where AC fails.

No complete Mammen spaces with countable universe under Lebesgue or Baire measurability.

Cardinal invariants $al u_M$ and $al u_T$ are characterized under different set-theoretic assumptions.

Abstract

We investigate the mathematics of a model of the human mind which has been proposed by the psychologist Jens Mammen. Mathematical realizations of this model consist of so-called \emph{Mammen spaces}, where a Mammen space is a triple $(U,\mathcal S,\mathcal C)$, where $U$ is a non-empty set ("the universe"), $\mathcal S$ is a perfect Hausdorff topology on $U$, and $\mathcal C\subseteq\mathcal P(U)$ together with $\mathcal S$ satisfy certain axioms. We refute a conjecture put forward by J. Hoffmann-J{\o}rgensen, who conjectured that the existence of a "complete" Mammen space implies the Axiom of Choice, by showing that in the first Cohen model, in which ZF holds but AC fails, there is a complete Mammen space. We obtain this by proving that in the first Cohen model, every perfect topology can be extended to a maximal perfect topology. On the other hand, we also show that if all sets are Lebesgue measurable, or all sets are Baire measurable, then there are no complete Mammen spaces with a countable universe. Finally, we investigate two new cardinal invariants $\mathfrak u_M$ and $\mathfrak u_T$ associated with complete Mammen spaces and maximal perfect topologies, and establish some basic inequalities that are provable in ZFC. We show $\mathfrak u_M=\mathfrak u_T=2^{\aleph_0}$ follows from Martin's Axiom, and, contrastingly, we show that $\aleph_1=\mathfrak u_M=\mathfrak u_T<2^{\aleph_0}=\aleph_2$ in the Baumgartner-Laver model.