Set theory and a model of the mind in psychology

Research output: Contribution to journal › Journal article › Research › peer-review

Standard

Set theory and a model of the mind in psychology. / Tornquist, Asger; Mammen, Jens.

In: Review of Symbolic Logic, Vol. 16, No. 4, 2023, p. 1233-1259.

Research output: Contribution to journal › Journal article › Research › peer-review

Harvard

Tornquist, A & Mammen, J 2023, 'Set theory and a model of the mind in psychology', Review of Symbolic Logic, vol. 16, no. 4, pp. 1233-1259. https://doi.org/10.1017/S1755020322000107

APA

Tornquist, A., & Mammen, J. (2023). Set theory and a model of the mind in psychology. Review of Symbolic Logic, 16(4), 1233-1259. https://doi.org/10.1017/S1755020322000107

Vancouver

Tornquist A, Mammen J. Set theory and a model of the mind in psychology. Review of Symbolic Logic. 2023;16(4):1233-1259. https://doi.org/10.1017/S1755020322000107

Author

Tornquist, Asger ; Mammen, Jens. / Set theory and a model of the mind in psychology. In: Review of Symbolic Logic. 2023 ; Vol. 16, No. 4. pp. 1233-1259.

Bibtex

@article{7eca3af0d1f543ac8718da53250d6ae5,
title = "Set theory and a model of the mind in psychology",
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 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-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. Further, we investigate two new cardinal invariants uM and uT associated with complete Mammen spaces and maximal perfect topologies, and establish some basic inequalities that are provable in ZFC. Then we show uM = uT = 2ℵ0 follows from Martin{\textquoteright}s Axiom, and, contrastingly, we show that ℵ1 = uM = uT < 2ℵ0 = ℵ2 in the Baumgartner-Laver model.",
author = "Asger Tornquist and Jens Mammen",
note = "Publisher Copyright: {\textcopyright} 2022 Cambridge University Press. All rights reserved.",
year = "2023",
doi = "10.1017/S1755020322000107",
language = "English",
volume = "16",
pages = "1233--1259",
journal = "Review of Symbolic Logic",
issn = "1755-0203",
publisher = "Cambridge University Press",
number = "4",

}

RIS

TY - JOUR

T1 - Set theory and a model of the mind in psychology

AU - Tornquist, Asger

AU - Mammen, Jens

N1 - Publisher Copyright: © 2022 Cambridge University Press. All rights reserved.

PY - 2023

Y1 - 2023

N2 - 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 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 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 complete Mammen 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.

AB - 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 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 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 complete Mammen 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.

U2 - 10.1017/S1755020322000107

DO - 10.1017/S1755020322000107

M3 - Journal article

AN - SCOPUS:85127139171

VL - 16

SP - 1233

EP - 1259

JO - Review of Symbolic Logic

JF - Review of Symbolic Logic

SN - 1755-0203

IS - 4

ER -

ID: 310562202