This paper can be seen as a mathematical elaboration of some fundamental open questions concerning the sufficiency or “completeness” of so-called sense and choice categories as basic presented in
Mammen, J. (2017). A New Logical Foundation for Psychology. Cham: Springer,
especially pp. 84-88, and a correction to their treatment in
Mammen, J. (2019). A grammar of praxis: An expose of “A new logical ´ foundation for psychology,” a few additions, and replies to Alaric Kohler and AlexanderPoddiakov. Integrative Psychological and Behavioral Science, 53(2),223–237.
At the same time the elaboration is an introduction to the extensional method bridging psychology and mathematical logic and a contribution to the logical foundation of mathematics itself.
An important conclusion is that completeness of sense and choice categories implies some non-constructive choice princple weaker than Axiom of Choice, and that sense categories in this case can't be arranged by any rule or algorithm ("constructively") and therefore can't be modeled by any computer, not even approximally.
Abstract (from the paper): 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 what the first author (A.T.) has 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 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 measurable, or all sets are Baire measurable, then there are no complete Mammen spaces with a countable universe. Further, we investigate two new cardinal invariants uM and uT associated with completeMammen spaces and maximal perfect topologies, and establish some basic inequalities that are provable in ZFC. Then we show uM = uT = 2ℵ0 follows from Martin’s Axiom, and, contrastingly, we show that ℵ1 = uM = uT < 2ℵ0 = ℵ2 in the Baumgartner–Laver model. Finally, consequences for psychology are discussed.