, 25:281 | Cite as

Continuous Fraïssé Conjecture

  • Arnold Beckmann
  • Martin GoldsternEmail author
  • Norbert Preining


We will investigate the relation of countable closed linear orderings with respect to continuous monotone embeddability and will show that there are exactly \(\aleph_1\) many equivalence classes with respect to this embeddability relation. This is an extension of Laver’s result (Laver, Ann. Math. 93(2):89–111, 1971), who considered (plain) embeddability, which yields coarser equivalence classes. Using this result we show that there are only \(\aleph_0\) many different Gödel logics.


Better quasi order Gödel logic Wellquasiorder Fraisse conjecture Continuous embeddings 


  1. 1.
    Baaz, M.: Infinite-valued Gödel logics with 0-1-projections and relativizations. In: Hájek, P., (ed.) Proc. Gödel‘96, Logic Foundations of Mathematics, Computer Science and Physics – Kurt Gödel’s Legacy. Lecture Notes in Logic 6, pp. 23–33. Springer, New York (1996)Google Scholar
  2. 2.
    Baaz, M., Leitsch, A., Zach, R.: Completeness of a first-order temporal logic with time-gaps. Theor. Comput. Sci. 160(1–2), 241–270 June (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Dushnik, B., Miller, E.W.: Concerning similarity transformations of linearly ordered sets. Bull. Am. Math. Soc. 46, 322–326 (1940)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Dunn, J.M., Meyer, R.K.: Algebraic completeness results for Dummett’s LC and its extensions. Z. Math. Log. Grundl. Math. 17, 225–230 (1971)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dummett, M.: A propositional logic with denumerable matrix. J. Symb. Log. 24, 96–107 (1959)MathSciNetGoogle Scholar
  6. 6.
    Erdős, P., Hajnal, A., Máté, A., Rado, R.: Combinatorial Set Theory: Partition Relations for Cardinals. Studies in Logic and the Foundations of Mathematics, vol. 106. North-Holland, Amsterdam (1984)Google Scholar
  7. 7.
    Fraïssé, R.: Sur la comparaison des types d’ordres. C. R. Acad. Sci. Paris 226, 1330–1331 (1948)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Gödel, K.: Zum Intuitionistischen Aussagenkalkül. Ergebnisse Eines Mathematischen Kolloquiums 4, 34–38 (1933)Google Scholar
  9. 9.
    Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Deventer (1998)zbMATHGoogle Scholar
  10. 10.
    Horn, A.: Logic with truth values in a linearly ordered Heyting algebra. J. Symb. Log. 34(3), 395–409 (1969)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Laver, R.: On Fraïssé’s order type conjecture. Ann. Math. 93(2), 89–111 (1971)MathSciNetGoogle Scholar
  12. 12.
    Mansfield, R.: Perfect subsets of definable sets of real numbers. Pac. J. Math. 35, 451–457 (1970)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Moschovakis, Y.N.: Descriptive Set Theory. Volume 100 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam (1980)Google Scholar
  14. 14.
    Minari, P., Takano, M., Ono, H.: Intermediate predicate logics determined by ordinals. J. Symb. Log. 55(3), 1099–1124 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Nash-Williams, C.St.J.A.: On well-quasi-ordering finite trees. Proc. Camb. Philos. Soc. 59, 833–835 (1963)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Nash-Williams, C.St.J.A.: On well-quasi-ordering lower sets of finite trees. Proc. Camb. Philos. Soc. 60, 369–384 (1964)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Nash-Williams, C.St.J.A.: On well-quasi-ordering infinite trees. Proc. Camb. Philos. Soc. 61, 697–720 (1965)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Nash-Williams, C.St.J.A.: On well-quasi-ordering transfinite sequences. Proc. Camb. Philos. Soc. 61, 33–39 (1965)zbMATHCrossRefGoogle Scholar
  19. 19.
    Nash-Williams, C.St.J.A.: On better-quasi-ordering transfinite sequences. Proc. Camb. Philos. Soc. 64, 273–290 (1968)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Preining, N.: Gödel logics and Cantor-Bendixon analysis. In: Baaz, M., Voronkov, A. (eds.) Proceedings of lpar ‘2002. lnai 2514, pp. 327–336, Tbilisi, 14–18 October 2002Google Scholar
  21. 21.
    Preining, N.: Complete Recursive Axiomatizability of Gödel Logics. PhD thesis, Vienna University of Technology, Austria (2003)Google Scholar
  22. 22.
    Rosenstein, J.G.: Linear Orderings. Volume 98 of Pure and Applied Mathematics. Academic [Harcourt Brace Jovanovich], New York (1982)Google Scholar
  23. 23.
    Scarpellini, B.: Die Nichtaxiomatisierbarkeit des unendlichwertigen Prädikatenkalküls von Łukasiewicz. J. Symb. Log. 27, 159–170 (1962)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Sierpiński, W.,: Sur les types d’ordre des ensembles linéaires. Fundam. Math. 37, 253–264 (1950)zbMATHGoogle Scholar
  25. 25.
    Takeuti, G., Titani, T.: Intuitionistic fuzzy logic and intuitionistic fuzzy set theory. J. Symb. Log. 49, 851–866 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Visser, A.: On the completeness principle: a study of provability in Heyting’s arithmetic. Ann. Math. Logic 22, 263–295 (1982)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  • Arnold Beckmann
    • 1
  • Martin Goldstern
    • 2
    Email author
  • Norbert Preining
    • 2
  1. 1.Department of Computer ScienceSwansea UniversitySwanseaUK
  2. 2.Institute of Discrete Mathematics and GeometryVienna University of TechnologyWienAustria

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