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SCOTT COMPLEXITY OF COUNTABLE STRUCTURES

Published online by Cambridge University Press:  01 February 2021

RACHAEL ALVIR
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NOTRE DAMESOUTH BEND, IN, USAE-mail: Rachael.C.Alvir.1@nd.edu
NOAM GREENBERG
Affiliation:
SCHOOL OF MATHEMATICS AND STATISTICS VICTORIA UNIVERSITY OF WELLINGTONWELLINGTON, NEW ZEALANDE-mail: greenberg@msor.vuw.ac.nzE-mail: dan.turetsky@vuw.ac.nz
DAN TURETSKY
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF MICHIGANANN ARBOR, MI, USAE-mail: matthhar@umich.edu

Abstract

We define the Scott complexity of a countable structure to be the least complexity of a Scott sentence for that structure. This is a finer notion of complexity than Scott rank: it distinguishes between whether the simplest Scott sentence is $\Sigma _{\alpha }$ , $\Pi _{\alpha }$ , or $\mathrm {d-}\Sigma _{\alpha }$ . We give a complete classification of the possible Scott complexities, including an example of a structure whose simplest Scott sentence is $\Sigma _{\lambda + 1}$ for $\lambda $ a limit ordinal. This answers a question left open by A. Miller.

We also construct examples of computable structures of high Scott rank with Scott complexities $\Sigma _{\omega _1^{CK}+1}$ and $\mathrm {d-}\Sigma _{\omega _1^{CK}+1}$ . There are three other possible Scott complexities for a computable structure of high Scott rank: $\Pi _{\omega _1^{CK}}$ , $\Pi _{\omega _1^{CK}+1}$ , $\Sigma _{\omega _1^{CK}+1}$ . Examples of these were already known. Our examples are computable structures of Scott rank $\omega _1^{CK}+1$ which, after naming finitely many constants, have Scott rank $\omega _1^{CK}$ . The existence of such structures was an open question.

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Article
Copyright
© Association for Symbolic Logic 2021

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