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 Algebra Logika: Year: Volume: Issue: Page: Find

 Algebra Logika, 2015, Volume 54, Number 5, Pages 551–574 (Mi al712)

Ash’s theorem on $\Delta^0_\alpha$-categorical structures and a condition for infinite $\Delta^0_\alpha$-dimension

P. E. Alaevab

a Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090, Russia
b Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia

Abstract: An old classical result in computable structure theory is Ash's theorem stating that for every computable ordinal $\alpha\ge2$, under some additional conditions, a computable structure is $\Delta^0_\alpha$-categorical iff it has a computable $\Sigma_\alpha$ Scott family. We construct a counterexample revealing that the proof of this theorem has a serious error. Moreover, we show how the error can be corrected by revising the proof. In addition, we formulate a sufficient condition under which the $\Delta^0_\alpha$-dimension of a computable structure is infinite.

Keywords: computable structure, Ash's theorem, $\Delta^0_\alpha$-categorical structure, $\Sigma_\alpha$ Scott family, $\Delta^0_\alpha$-dimension of a computable structure.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation ÍØ-860.2014.1 Russian Foundation for Basic Research 14-01-00376 Supported by the Grants Council (under RF President) for State Aid of Leading Scientific Schools (grant NSh-860.2014.1) and by RFBR (project No. 14-01-00376).

DOI: https://doi.org/10.17377/alglog.2015.54.501

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English version:
Algebra and Logic, 2015, 54:5, 353–369

Bibliographic databases:

UDC: 510.5+510.6

Citation: P. E. Alaev, “Ash’s theorem on $\Delta^0_\alpha$-categorical structures and a condition for infinite $\Delta^0_\alpha$-dimension”, Algebra Logika, 54:5 (2015), 551–574; Algebra and Logic, 54:5 (2015), 353–369

Citation in format AMSBIB
\Bibitem{Ala15} \by P.~E.~Alaev \paper Ash’s theorem on $\Delta^0_\alpha$-categorical structures and a~condition for infinite $\Delta^0_\alpha$-dimension \jour Algebra Logika \yr 2015 \vol 54 \issue 5 \pages 551--574 \mathnet{http://mi.mathnet.ru/al712} \crossref{https://doi.org/10.17377/alglog.2015.54.501} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3468417} \transl \jour Algebra and Logic \yr 2015 \vol 54 \issue 5 \pages 353--369 \crossref{https://doi.org/10.1007/s10469-015-9357-2} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000366155000001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84957939091}