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Sibirsk. Mat. Zh., 2011, Volume 52, Number 6, Pages 1199–1220 (Mi smj2268)  

This article is cited in 2 scientific papers (total in 2 papers)

On the admissible sets of type $\mathbb{HYP}(\mathfrak M)$ over recursively saturated models

R. R. Avdeev

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk

Abstract: Some effective expression is obtained for the elements of an admissible set $\mathbb{HYP}(\mathfrak M)$ as template sets. We prove the $\Sigma$-reducibility of $\mathbb{HYP}(\mathfrak M)$ to $\mathbb{HF}(\mathfrak M)$ for each recursively saturated model $\mathfrak M$ of a regular theory, give a criterion for uniformization in $\mathbb{HYP}(\mathfrak M)$ for each recursively saturated model $\mathfrak M$, and establish uniformization in $\mathbb{HYP}(\mathfrak N)$ and $\mathbb{HYP}(\mathfrak R')$, where $\mathfrak N$ and $\mathfrak R'$ are recursively saturated models of arithmetic and real closed fields. We also prove the absence of uniformization in $\mathbb{HF}(\mathfrak M)$ and $\mathbb{HYP}(\mathfrak M)$ for each countably saturated model $\mathfrak M$ of an uncountably categorical theory, and give an example of this type of theory with definable Skolem functions. Furthermore, some example is given of a model of a regular theory with $\Sigma$-definable Skolem functions, but lacking definable Skolem functions in every extension by finitely many constants.

Keywords: admissible set, HYP, HF, recursively saturated model, uniformization, template set, $\Sigma$-reducibility, Skolem functions.

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English version:
Siberian Mathematical Journal, 2011, 52:6, 951–968

Bibliographic databases:

UDC: 510.5
Received: 27.06.2010
Revised: 07.06.2011

Citation: R. R. Avdeev, “On the admissible sets of type $\mathbb{HYP}(\mathfrak M)$ over recursively saturated models”, Sibirsk. Mat. Zh., 52:6 (2011), 1199–1220; Siberian Math. J., 52:6 (2011), 951–968

Citation in format AMSBIB
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\by R.~R.~Avdeev
\paper On the admissible sets of type $\mathbb{HYP}(\mathfrak M)$ over recursively saturated models
\jour Sibirsk. Mat. Zh.
\yr 2011
\vol 52
\issue 6
\pages 1199--1220
\mathnet{http://mi.mathnet.ru/smj2268}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2961749}
\transl
\jour Siberian Math. J.
\yr 2011
\vol 52
\issue 6
\pages 951--968
\crossref{https://doi.org/10.1134/S0037446611060012}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84855170940}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. R. R. Avdeev, V. G. Puzarenko, “A computable structure with nonstandard computability”, Algebra and Logic, 56:5 (2017), 425–426  mathnet  crossref  crossref  isi
    2. R. R. Avdeev, V. G. Puzarenko, “A computable structure with non-standard computability”, Siberian Adv. Math., 29:2 (2019), 77–115  mathnet  crossref  crossref
  • Сибирский математический журнал Siberian Mathematical Journal
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