Infinite Games (L24)

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Infinite Games (L24) Professor Benedikt Löwe Infinite two-player perfect information games are connected to many topics in the foundations of mathematics: central concepts from analysis and topology can be reformulated in game- theoretic terms using infinite two-player perfect information games. Examples are the concepts of Lebesgue measurability, the property of Baire, as well as the perfect set property. The central game-theoretic notion is the concept of determinacy: the full axiom of determinacy AD (\all infinite two-player perfect information games with natural number moves are deter- mined") contradicts the axiom of choice AC, but definable fragments of AD can be proved in ZFC or extensions of ZFC. The axiom of determinacy itself yields an interesting alternative foundations of mathematics. We shall treat several of the following topics: Basic theory of determinacy. Applications in topology and measure theory. Incompatibility of AC and AD. Basics of descriptive set theory: the Borel hierarchy and the projective hierarchy. Proving determinacy. Open determinacy. Low-level Borel games. Borel determinacy. Proving determinacy from large cardinals. Introduction to large cardinals: inaccessible 1 cardinals and measurable cardinals. Proving Π1-determinacy from a measurable cardinal. The axiom of determinacy. Combinatorial consequences: @1 and @2 are measurable. Infinite exponent partition relations. Stronger axioms of determinacy. The axiom of real determinacy. Inconsistent extensions of the axiom of determinacy. Long games. The Wadge hierarchy. Definition and structure theory of the Wadge hierarchy under AD. A course webpage will be available at https://www.math.uni-hamburg.de/home/loewe/Lent2020/. Pre-requisites The Part II course Logic and Set Theory or an equivalent course is essential. The Part Ib course Metric and Topological Spaces is useful. Literature 1. Akihiro Kanamori, The Higher Infinite. Large Cardinals in Set Theory from Their Be- ginnings. Springer 2003 [Springer Monographs in Mathematics] Additional support Four examples sheets will be provided and four associated examples classes will be given. There will be a revision class in the Easter Term. 1.

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The Wadge Hierarchy : Beyond Borel Sets
Unicentre CH-1015 Lausanne http://serval.unil.ch Year : 2016 The Wadge Hierarchy : Beyond Borel Sets FOURNIER Kevin FOURNIER Kevin, 2016, The Wadge Hierarchy : Beyond Borel Sets Originally published at : Thesis, University of Lausanne Posted at the University of Lausanne Open Archive http://serval.unil.ch Document URN : urn:nbn:ch:serval-BIB_B205A3947C037 Droits d’auteur L'Université de Lausanne attire expressément l'attention des utilisateurs sur le fait que tous les documents publiés dans l'Archive SERVAL sont protégés par le droit d'auteur, conformément à la loi fédérale sur le droit d'auteur et les droits voisins (LDA). A ce titre, il est indispensable d'obtenir le consentement préalable de l'auteur et/ou de l’éditeur avant toute utilisation d'une oeuvre ou d'une partie d'une oeuvre ne relevant pas d'une utilisation à des fins personnelles au sens de la LDA (art. 19, al. 1 lettre a). A défaut, tout contrevenant s'expose aux sanctions prévues par cette loi. Nous déclinons toute responsabilité en la matière. Copyright The University of Lausanne expressly draws the attention of users to the fact that all documents published in the SERVAL Archive are protected by copyright in accordance with federal law on copyright and similar rights (LDA). Accordingly it is indispensable to obtain prior consent from the author and/or publisher before any use of a work or part of a work for purposes other than personal use within the meaning of LDA (art. 19, para. 1 letter a). Failure to do so will expose offenders to the sanctions laid down by this law.
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