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Logic"],"published-print":{"date-parts":[[2025,2]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>In the context of a weak formal theory called Basic Intuitionistic Mathematics <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\textsf{BIM}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>BIM<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, we study Brouwer\u2019s <jats:italic>Fan Theorem<\/jats:italic> and a strong negation of the Fan Theorem, <jats:italic>Kleene\u2019s Alternative (to the Fan Theorem)<\/jats:italic>. We prove that the Fan Theorem is equivalent to <jats:italic>contrapositions<\/jats:italic> of a number of intuitionistically accepted axioms of countable choice and that Kleene\u2019s Alternative is equivalent to <jats:italic>strong negations<\/jats:italic> of these statements. We discuss finite and infinite games and introduce a constructively useful notion of <jats:italic>determinacy<\/jats:italic>. We prove that the Fan Theorem is equivalent to the <jats:italic>Intuitionistic Determinacy Theorem<\/jats:italic>. This theorem says that every subset of Cantor space <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$2^\\omega $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mn>2<\/mml:mn>\n                    <mml:mi>\u03c9<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> is, in our constructively meaningful sense, determinate. Kleene\u2019s Alternative is equivalent to a strong negation of a special case of this theorem. We also consider a <jats:italic>uniform intermediate value theorem<\/jats:italic> and a <jats:italic>compactness theorem for classical propositional logic<\/jats:italic>. The Fan Theorem is equivalent to each of these theorems and Kleene\u2019s Alternative is equivalent to strong negations of them. We end with a note on <jats:italic>\u2018stronger\u2019<\/jats:italic> Fan Theorems. The paper is a sequel to Veldman (Arch Math Logic 53:621\u2013693, 2014).<\/jats:p>","DOI":"10.1007\/s00153-024-00930-9","type":"journal-article","created":{"date-parts":[[2024,6,6]],"date-time":"2024-06-06T20:17:31Z","timestamp":1717705051000},"page":"1-66","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Fan Theorem, its strong negation, and the determinacy of games"],"prefix":"10.1007","volume":"64","author":[{"given":"Wim","family":"Veldman","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,6,6]]},"reference":[{"key":"930_CR1","doi-asserted-by":"crossref","unstructured":"Akama, Y., Berardi, S., Hayashi, S., Kohlenbach, U.: An arithmetical hierarchy of the law of excluded middle and related principles. 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