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 and an ordinal parameter, that 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 further define the recognizable closure for subsets of λ by closing under relative recognizability for subsets of λ.

We prove several results about recognizable sets and their variants for other types of machines. Notably, we show the following results from large cardinals.

  • Recognizable sets of ordinals appear in the hierarchy of inner models at least up to the level Woodin cardinals, while computable sets are elements of L.

  • A subset of a countable ordinal λ is in the recognizable closure for subsets of countable ordinals if and only if it is an element of the inner model M, which is obtained by iterating the least measure of the least fine structural inner model M1 with a Woodin cardinal through the ordinals.

MSC

03D10
03D32
03E15
03E45

Keywords

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

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Work on this paper was partially done whilst the authors were visiting at the Isaac Newton Institute for Mathematical Sciences in the programme ‘Mathematical, Foundational and Computational Aspects of the Higher Infinite’ (HIF), to which they are grateful. This program was funded by EPSRC grant EP/K032208/1. In addition the second author was partially supported by DFG-grant LU2020/1-1. Moreover, the last author was a Simons Foundation Fellow during this period and gratefully acknowledges that Foundation's support.