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Mat. Sb., 2012, Volume 203, Number 8, Pages 141–160 (Mi msb7883)  

This article is cited in 3 scientific papers (total in 3 papers)

Undecidability of the elementary theory of the semilattice of GLP-words

F. N. Pakhomov

M. V. Lomonosov Moscow State University

Abstract: The Lindenbaum algebra of Peano PA can be enriched by the $n$-consistency operators which assign, to a given formula, the statement that the formula is compatible with the theory PA extended by the set of all true $\Pi_n$-sentences. In the Lindenbaum algebra of PA, a lower semilattice is generated from $\mathbf{1}$ by the $n$-consistency operators. We prove the undecidability of the elementary theory of this semilattice and the decidability of the elementary theory of the subsemilattice (of this semilattice) generated by the $0$-consistency and $1$-consistency operators only.
Bibliography: 16 titles.

Keywords: provability logic, elementary theories, undecidability.

DOI: https://doi.org/10.4213/sm7883

Full text: PDF file (547 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2012, 203:8, 1211–1229

Bibliographic databases:

UDC: 510.673
MSC: Primary 03F45; Secondary 03B25, 03F25
Received: 02.05.2011

Citation: F. N. Pakhomov, “Undecidability of the elementary theory of the semilattice of GLP-words”, Mat. Sb., 203:8 (2012), 141–160; Sb. Math., 203:8 (2012), 1211–1229

Citation in format AMSBIB
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  • https://doi.org/10.4213/sm7883
  • http://mi.mathnet.ru/eng/msb/v203/i8/p141

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. F. N. Pakhomov, “On elementary theories of ordinal notation systems based on reflection principles”, Proc. Steklov Inst. Math., 289 (2015), 194–212  mathnet  crossref  crossref  isi  elib
    2. F. N. Pakhomov, “Linear $\mathrm{GLP}$-algebras and their elementary theories”, Izv. Math., 80:6 (2016), 1159–1199  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. L. Beklemishev, T. Flaminio, “Franco Montagna's work on provability logic and many-valued logic”, Studia Logica, 104:1 (2016), 1–46  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник Sbornik: Mathematics (from 1967)
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