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Recognizable sets and Woodin cardinals : Computation beyond the constructible universe

Recognizable sets and Woodin cardinals : Computation beyond the constructible universe

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CARL, Merlin, Philipp SCHLICHT, Philip WELCH, 2015. Recognizable sets and Woodin cardinals : Computation beyond the constructible universe

@unpublished{Carl2015Recog-32698, title={Recognizable sets and Woodin cardinals : Computation beyond the constructible universe}, year={2015}, author={Carl, Merlin and Schlicht, Philipp and Welch, Philip} }

2016-01-25T13:57:54Z Recognizable sets and Woodin cardinals : Computation beyond the constructible universe 2016-01-25T13:57:54Z eng Welch, Philip Schlicht, Philipp Welch, Philip Carl, Merlin We call a subset of an ordinal λ recognizable if it is the unique subset x of λ for which some Turing machine with ordinal time and tape, which halts for all subsets of λ as input, halts with the final state 0. Equivalently, such a set is the unique subset x which satisfies a given Σ1 formula in L[x]. We prove several results about sets of ordinals recognizable from ordinal parameters by ordinal time Turing machines. Notably we show the following results from large cardinals.<br />(1) Computable sets are elements of L, while recognizable objects with infinite time computations appear up to the level of Woodin cardinals.<br />(2) A subset of a countable ordinal λ is in the recognizable closure for subsets of λ if and only if it is an element of M∞, where M∞ denotes the inner model obtained by iterating the least measure of M1 through the ordinals, and where the recognizable closure for subsets of λ is defined by closing under relative recognizability for subsets of λ. Schlicht, Philipp 2015 Carl, Merlin

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