Incompleteness of a first-order gödel logic and some temporal logics of programs

  • Matthias Baaz
  • Alexander Leitsch
  • Richard Zach
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1092)


It is shown that the infinite-valued first-order Gödel logic G° based on the set of truth values {1/k: k ε w {0}} U {0} is not r.e. The logic G° is the same as that obtained from the Kripke semantics for first-order intuitionistic logic with constant domains and where the order structure of the model is linear. From this, the unaxiomatizability of Kröger's temporal logic of programs (even of the fragment without the nexttime operator O) and of the authors' temporal logic of linear discrete time with gaps follows.


temporal logic intermediate logic many-valued logic 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Baaz, C. G. Fermüller, and R. Zach. Elimination of cuts in first-order finite-valued logics. J. Inform. Process. Cybernet. EIK, 29(6), 333–355, 1994.Google Scholar
  2. 2.
    M. Baaz, A. Leitsch, and R. Zach. Completeness of a first-order temporal logic with time-gaps. Theoret. Comput. Sci., 1996 to appear.Google Scholar
  3. 3.
    M. Dummett. A prepositional calculus with denumerable matrix. J. Symbolic Logic, 24, 97–106, 1959.Google Scholar
  4. 4.
    K. Gödel. Zum intuitionistischen Aussagenkalkül. Anz. Akad. Wiss. Wien, 69, 65–66, 1932.Google Scholar
  5. 5.
    F. Kroger. Temporal Logic of Programs. EATCS Monographs in Computer Science 8. (Springer, Berlin, 1987).Google Scholar
  6. 6.
    F. Kröger. On the interpretability of arithmetic in temporal logic. Theoret. Cornput. Sci., 73, 47–60, 1990.Google Scholar
  7. 7.
    S. Merz. Decidability and incompleteness results for first-order temporal logics of linear time. J. Applied Non-Classical Logics, 2(2), 139–156, 1992.Google Scholar
  8. 8.
    B. Scarpellini. Die Nichtaxiomatisierbarkeit des unendlichwertigen Prädikatenkalküls von Lukasiewicz. J. Symbolic Logic, 27, 159–170, 1962.Google Scholar
  9. 9.
    A. Szalas. Concerning the semantic consequence relation in first-order temporal logic. Theoret. Comput. Sci., 47, 329–334, 1986.Google Scholar
  10. 10.
    A. Szalas and L. Holenderski. Incompleteness of first-order temporal logic with until. Theoret. Comput. Sci., 57, 317–325, 1988.Google Scholar
  11. 11.
    G. Takeuti and T. Titani. Intuitionistic fuzzy logic and instuitionistic fuzzy set theory. J. Symbolic Logic, 49, 851–866, 1984.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Matthias Baaz
    • 1
  • Alexander Leitsch
    • 2
  • Richard Zach
    • 2
  1. 1.Institut für Algebra, und Diskrete Mathematik E118.2Technische Universität WienViennaAustria
  2. 2.Institut für Computersprachen E185.2Technische Universität WienViennaAustria

Personalised recommendations