Borel Determinacy and Metamathematics

M.A. in Mathematics, University of North Texas, 2001

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Abstract: Borel Determinacy (BD) states that in $G(T,X)$ is a game and $X$ is Borel then $G(T,X)$ is determined.  Proved by Martin in 1975, BD 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 ($/aleph_1$ many power sets of $/omega$).  Friedman proved in 1971 that these sets are necessary by showing that the Axiom of Replacement is necessary for any proof of BD.  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, BD is not a theorem of ZC.  This paper contains three main sections: Martin's proof of BD; 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.