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AN INTRODUCTION TO THE SCOTT COMPLEXITY OF COUNTABLE STRUCTURES AND A SURVEY OF RECENT RESULTS

Part of: Computability and recursion theory Model theory

Published online by Cambridge University Press:  15 November 2021

MATTHEW HARRISON-TRAINOR
MATTHEW HARRISON-TRAINOR*
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF MICHIGAN EAST HALL ANN ARBOR, MI 48109, USA E-mail: matthhar@umich.edu
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Abstract

Every countable structure has a sentence of the infinitary logic $\mathcal {L}_{\omega _1 \omega }$ which characterizes that structure up to isomorphism among countable structures. Such a sentence is called a Scott sentence, and can be thought of as a description of the structure. The least complexity of a Scott sentence for a structure can be thought of as a measurement of the complexity of describing the structure. We begin with an introduction to the area, with short and simple proofs where possible, followed by a survey of recent advances.

Keywords

computable structure theoryScott rank

MSC classification

Primary: 03D45: Theory of numerations, effectively presented structures
Secondary: 03C57: Effective and recursion-theoretic model theory 03C70: Logic on admissible sets

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Type
Articles
Information
Bulletin of Symbolic Logic , Volume 28 , Issue 1 , March 2022 , pp. 71 - 103
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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