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1	Borel Determinacy and Metamathematics Bryant, Ross 12 1900 (has links) Borel determinacy states that if G(T;X) is a game and X is Borel, then G(T;X) is determined. Proved by Martin in 1975, Borel determinacy is a theorem of ZFC set theory, and is, in fact, the best determinacy result in ZFC. However, the proof uses sets of high set theoretic type (N1 many power sets of ω). Friedman proved in 1971 that these sets are necessary by showing that the Axiom of Replacement is necessary for any proof of Borel Determinacy. To prove this, Friedman produces a model of ZC and a Borel set of Turing degrees that neither contains nor omits a cone; so by another theorem of Martin, Borel Determinacy is not a theorem of ZC. This paper contains three main sections: Martin's proof of Borel Determinacy; a simpler example of Friedman's result, namely, (in ZFC) a coanalytic set of Turing degrees that neither contains nor omits a cone; and finally, the Friedman result. Descriptive set theory. Metamathematics. Borel Determinacy Descriptive Set Theory Logic Foundations
2	Determinacy in the Low Levels of the Projective Hierarchy Cotton, Michael R. 06 August 2012 (has links) No description available. Logic Mathematics determinacy descriptive set theory large cardinal measurable cardinal set theory projective Borel ZFC homogeneously Suslin ultrapower Ramsey cardinal analytic determinacy analytic Borel hierarchy projective hierarchy sets Borel determinacy