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Logic Seminar

Wednesday, January 19, 2022
12:00pm to 1:00pm
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Set theory and a proposed model of the mind in psychology
Asger Törnquist, Department of Mathematics, University of Copenhagen,

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.

For more information, please email A. Kechris at [email protected].