Recognizable sets and Woodin cardinals: computation beyond the constructible universe

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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.
Original languageEnglish
Pages (from-to)312-332
Number of pages21
JournalAnnals of Pure and Applied Logic
Volume169
Issue number4
Early online date13 Dec 2017
DOIs
Publication statusPublished - Apr 2018

Keywords

  • Infinite time
  • Turing machines
  • Algorithmic randomness
  • Effective descriptive set theory
  • Woodin cardinals
  • Inner models

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