838 research outputs found

    Infinite time Turing machines and an application to the hierarchy of equivalence relations on the reals

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    We describe the basic theory of infinite time Turing machines and some recent developments, including the infinite time degree theory, infinite time complexity theory, and infinite time computable model theory. We focus particularly on the application of infinite time Turing machines to the analysis of the hierarchy of equivalence relations on the reals, in analogy with the theory arising from Borel reducibility. We define a notion of infinite time reducibility, which lifts much of the Borel theory into the class Ī”21\bm{\Delta}^1_2 in a satisfying way.Comment: Submitted to the Effective Mathematics of the Uncountable Conference, 200

    The bounded proper forcing axiom and well orderings of the reals

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    We show that the bounded proper forcing axiom BPFA implies that there is a well-ordering of P(Ļ‰_1) which is Ī”_1 definable with parameter a subset of Ļ‰_1. Our proof shows that if BPFA holds then any inner model of the universe of sets that correctly computes N_2 and also satisfies BPFA must contain all subsets of Ļ‰_1. We show as applications how to build minimal models of BPFA and that BPFA implies that the decision problem for the HƤrtig quantifier is not lightface projective
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