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Recognizable sets and Woodin cardinals: computation beyond the constructible universe

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Recognizable sets and Woodin cardinals : computation beyond the constructible universe. / Carl, Merlin; Schlicht, Philipp; Welch, Philip D.

In: Annals of Pure and Applied Logic, Vol. 169, No. 4, 04.2018, p. 312-332.

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Carl, Merlin ; Schlicht, Philipp ; Welch, Philip D. / Recognizable sets and Woodin cardinals : computation beyond the constructible universe. In: Annals of Pure and Applied Logic. 2018 ; Vol. 169, No. 4. pp. 312-332.

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@article{a35b0656a7bd4ed0a8e12ffe0a085762,
title = "Recognizable sets and Woodin cardinals: computation beyond the constructible universe",
abstract = "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.(1) Computable sets are elements of L, while recognizable objects with infinite time computations appear up to the level of Woodin cardinals.(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.",
keywords = "Infinite time, Turing machines, Algorithmic randomness, Effective descriptive set theory, Woodin cardinals, Inner models",
author = "Merlin Carl and Philipp Schlicht and Welch, {Philip D}",
year = "2018",
month = "4",
doi = "10.1016/j.apal.2017.12.007",
language = "English",
volume = "169",
pages = "312--332",
journal = "Annals of Pure and Applied Logic",
issn = "0168-0072",
publisher = "Elsevier Masson SAS",
number = "4",

}

RIS - suitable for import to EndNote

TY - JOUR

T1 - Recognizable sets and Woodin cardinals

T2 - computation beyond the constructible universe

AU - Carl, Merlin

AU - Schlicht, Philipp

AU - Welch, Philip D

PY - 2018/4

Y1 - 2018/4

N2 - 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.(1) Computable sets are elements of L, while recognizable objects with infinite time computations appear up to the level of Woodin cardinals.(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.

AB - 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.(1) Computable sets are elements of L, while recognizable objects with infinite time computations appear up to the level of Woodin cardinals.(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.

KW - Infinite time

KW - Turing machines

KW - Algorithmic randomness

KW - Effective descriptive set theory

KW - Woodin cardinals

KW - Inner models

UR - https://arxiv.org/abs/1512.06101

U2 - 10.1016/j.apal.2017.12.007

DO - 10.1016/j.apal.2017.12.007

M3 - Article

VL - 169

SP - 312

EP - 332

JO - Annals of Pure and Applied Logic

JF - Annals of Pure and Applied Logic

SN - 0168-0072

IS - 4

ER -