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

Ashs 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 -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

Full text: PDF file (245 kB)
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English version:
Algebra and Logic, 2015, 54:5, 353–369

Bibliographic databases:

UDC: 510.5+510.6
Received: 08.02.2015

Citation: P. E. Alaev, “Ashs 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
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\by P.~E.~Alaev
\paper Ashs 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}
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