{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T22:04:26Z","timestamp":1747173866136,"version":"3.40.5"},"reference-count":30,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2022,3,14]],"date-time":"2022-03-14T00:00:00Z","timestamp":1647216000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2023,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>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 consists of what the first author (A.T.) has called <jats:italic>Mammen spaces<\/jats:italic>, where a Mammen space is a triple <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000107_inline1.png\"\/><jats:tex-math>\n$(U,\\mathcal S,\\mathcal C)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, where <jats:italic>U<\/jats:italic> is a non-empty set (\u201cthe universe\u201d), <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000107_inline2.png\"\/><jats:tex-math>\n$\\mathcal S$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is a perfect Hausdorff topology on <jats:italic>U<\/jats:italic>, and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000107_inline3.png\"\/><jats:tex-math>\n$\\mathcal C\\subseteq \\mathcal P(U)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> together with <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000107_inline4.png\"\/><jats:tex-math>\n$\\mathcal S$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> satisfy certain axioms.<\/jats:p><jats:p>We refute a conjecture put forward by Hoffmann-J\u00f8rgensen, who conjectured that the existence of a \u201ccomplete\u201d 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.<\/jats:p><jats:p>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.<\/jats:p><jats:p>Further, we investigate two new cardinal invariants <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000107_inline5.png\"\/><jats:tex-math>\n$\\mathfrak u_{M}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000107_inline6.png\"\/><jats:tex-math>\n$\\mathfrak u_{T}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> associated with complete Mammen spaces and maximal perfect topologies, and establish some basic inequalities that are provable in ZFC. Then we show <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000107_inline7.png\"\/><jats:tex-math>\n$\\mathfrak u_{M}=\\mathfrak u_{T}=2^{\\aleph _{0}}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> follows from Martin\u2019s Axiom, and, contrastingly, we show that <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S1755020322000107_inline8.png\"\/><jats:tex-math>\n$\\aleph _{1}=\\mathfrak u_{M}=\\mathfrak u_{T}&lt;2^{\\aleph _{0}}=\\aleph _{2}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> in the Baumgartner\u2013Laver model.<\/jats:p><jats:p>Finally, consequences for psychology are discussed.<\/jats:p>","DOI":"10.1017\/s1755020322000107","type":"journal-article","created":{"date-parts":[[2022,3,14]],"date-time":"2022-03-14T08:05:13Z","timestamp":1647245113000},"page":"1233-1259","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["SET THEORY AND A MODEL OF THE MIND IN PSYCHOLOGY"],"prefix":"10.1017","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8332-649X","authenticated-orcid":false,"given":"ASGER","family":"T\u00d6RNQUIST","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8867-5744","authenticated-orcid":false,"given":"JENS","family":"MAMMEN","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2022,3,14]]},"reference":[{"key":"S1755020322000107_r29","doi-asserted-by":"publisher","DOI":"10.1016\/0166-8641(93)90145-4"},{"volume-title":"The Axiom of Choice","year":"1973","author":"Jech","key":"S1755020322000107_r11"},{"key":"S1755020322000107_r8","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-43-01029-4"},{"volume-title":"On Random Variability of Responses: A Note on Jens Mammen\u2019s Book","year":"2017","author":"Dzhafarov","key":"S1755020322000107_r3"},{"key":"S1755020322000107_r21","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-77892-7_34"},{"volume-title":"Individuals","year":"1964","author":"Strawson","key":"S1755020322000107_r28"},{"volume-title":"Sanse- og udvalgskategorielle strukturer i den skizofrene erkendelse\u2014og i psykiatriens [Sense and choice categorial structures in schizophrenic cognition\u2014and in psychiatry\u2019s]","year":"1990","author":"Mogensen","key":"S1755020322000107_r23"},{"key":"S1755020322000107_r20","doi-asserted-by":"publisher","DOI":"10.1007\/s12124-019-09487-2"},{"key":"S1755020322000107_r17","first-page":"29","volume-title":"The Societal Subject","author":"Mammen","year":"1993"},{"key":"S1755020322000107_r1","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(79)90010-X"},{"volume-title":"Problems of the Development of the Mind","year":"1981","author":"Leontiev","key":"S1755020322000107_r16"},{"key":"S1755020322000107_r2","doi-asserted-by":"publisher","DOI":"10.2307\/2586632"},{"key":"S1755020322000107_r7","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/013.1\/0284328"},{"key":"S1755020322000107_r13","first-page":"89","volume-title":"Human Characteristics","author":"Kr\u00f8jgaard","year":"2009"},{"key":"S1755020322000107_r14","doi-asserted-by":"publisher","DOI":"10.1007\/s12124-015-9340-4"},{"key":"S1755020322000107_r19","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-67783-5"},{"key":"S1755020322000107_r27","doi-asserted-by":"publisher","DOI":"10.2307\/1970696"},{"key":"S1755020322000107_r4","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-51088-0"},{"volume-title":"The Ecological Approach to Visual Perception","year":"1979","author":"Gibson","key":"S1755020322000107_r6"},{"key":"S1755020322000107_r24","doi-asserted-by":"publisher","DOI":"10.1007\/s12124-015-9332-4"},{"volume-title":"Den menneskelige sans. 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