arXiv:2109.06739v1 [math.LO] 14 Sep 2021 SET THEORY AND A MODEL OF THE MIND IN PSYCHOLOGY JENS MAMMEN, ASGER T ¨ ORNQUIST Abstract. We investigate the mathematics of a model of the human mind which has been proposed by the psychologist Jens Mammen. Math- ematical realizations of this model consist of so-called Mammen spaces, where a Mammen space is a triple (U, S , C), where U is a non-empty set (“the universe”), S is a perfect Hausdorff topology on U , and C⊆P (U ) together with S satisfy certain axioms. We refute a conjecture put forward by J. Hoffmann-Jørgensen, 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 mea- surable, or all sets are Baire measurable, then there are no complete Mammen spaces with a countable universe. Finally, we investigate two new cardinal invariants uM and uT asso- ciated with complete Mammen spaces and maximal perfect topologies, and establish some basic inequalities that are provable in ZFC. Further, we show uM = uT =2 ℵ 0 follows from Martin’s Axiom, and, contrast- ingly, we show that ℵ1 = uM = uT < 2 ℵ 0 = ℵ2 in the Baumgartner- Laver model. 1. Introduction In theoretical psychology, Jens Mammen has proposed a model for what may be called the interface between the inner world of a human mind, and the outer world that this human mind lives in, perceives, and interacts with. From the outset, Mammen has formulated and presented his theory axiomatically, in the style familiar to mathematicians. The purpose of this paper is to study the set-theoretic aspects of Mammen’s theory. Briefly, a Mammen space can be defined as follows: Definition 1.1. A Mammen space is a triple (U, S , C ), where U is a non- empty set, called the universe of objects, and S , C⊆P (U ) such that Date : Wednesday 15 th September, 2021. 2020 Mathematics Subject Classification. 03E05, 03E15, 03E17, 03E35, 03E45, 03E50, 54A10, 91E30. 1