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Theory of Linear Order in Extended Logics

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Quantifiers: Logics, Models and Computation

Part of the book series: Synthese Library ((SYLI,volume 248))

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Abstract

The notion of linear ordering is one of the fundamental concepts in mathematics. Linear orderings play an important role in algebra, topology, set theory and model theory. The present paper surveys contributions to the meta-theory and model theory of linear orderings. The framework for the considered problems is defined by logics L extending first order logic and satisfying reasonable model theoretic properties. The investigation is based on logics L(Q α ) with cardinality quantifiers Q α , logics L(Q m α ) with Malitz quantifiers Q m α , m > 1, and stationary logic L(aa). The paper is mainly concerned with the decision problem of linear orderings and the problem of weak classification by constructing subclasses of linear orderings that are in the topological sense dense with respect to the considered theories.

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References

  1. U. Abraham and S. Shelah, Isomorphism types of Aronszajn trees, Israel Journal of Mathematics 50 (1985), pp. 75–113.

    Article  Google Scholar 

  2. L. W. Badger, An Ehrenfeucht game for the multivariable quantifiers of Malitz and some applications, Pacific Journal of Mathematics 72 (1977), pp. 293–304.

    Article  Google Scholar 

  3. J. Baumgartner, A new class of order types, Annals of Math. Logic 9 (1976), pp. 187–222.

    Google Scholar 

  4. K. J. Barwise (ed.), Handbook of Mathematical Logic, North-Holland Publ. Company, 1977.

    Google Scholar 

  5. J. K. Barwise and S. Feferman, Model-Theoretic Logics, New York, Springer-Verlag, 1985.

    Google Scholar 

  6. J. Barwise, M. Kaufmann and M. Makkai, Stationary logic, Ann. Math. Logic 13 (1978), pp. 171–224.

    Article  Google Scholar 

  7. A. Baudisch, D. Seese, H.-P. Tuschik and M. Weese, Decidability and Generalized Quantifiers Akademie-Verlag, Berlin 1980, xii + 235pp.

    Google Scholar 

  8. J. R. Büchi, The monadic second order theory of wl,in:Decidable Theories II, LNM vol. 328 (1973), pp. 129–217.

    Google Scholar 

  9. K. J. Devlin and H. Johnsbraten, The Souslin Prob- lem, Lecture Notes in Math., vol. 405 (1975), Springer-Verlag, New-York.

    Google Scholar 

  10. P. C. Eklof and A.H. Mekler, Stationary logic of finitely determinate structures, Ann. Math. Logic 17 (1979), pp. 227–270.

    Article  Google Scholar 

  11. P. Erdös and A. Hajnal, On a classification of countable order types and an application to the partition calculus, Fund. Math. 51 (1962), pp. 117–129.

    Google Scholar 

  12. P.Erdös and R. Rado, A partition calculus in set theory, Bull. Amer. Math. Soc. 62 (1956), pp. 427–486.

    Google Scholar 

  13. H. Gaifman and E. P. Specker, Isomorphism types of trees,Proc. Amer. Math. Soc. 15 (1964), pp. 1–7.

    Google Scholar 

  14. S. C. Garavaglia, Relative strength of Malitz quantifiers, Notre Dame Journal of Formal Logic 19 (1978), pp. 495–503.

    Google Scholar 

  15. H.-J. Goltz, The Boolean sentence algebra of the theory of linear ordering is atomic with respect to logics with a Malitz quantifier, Zeitschr. f. math. Logik and Grundlagen d. Math., Bd. 31 (1985), pp. 131–162.

    Google Scholar 

  16. V. Harnik and M. Makkai, New ariomatizations for logics with generalized quantifiers, Isreael Journal of Mathematics 32 (1979), pp. 257–281.

    Article  Google Scholar 

  17. F. Hausdorff, Grundzuge einer Theorie der geordneten Mengen, Math. Annalen 65 (1908), pp. 435–505.

    Google Scholar 

  18. H. Herre, Miscellanous results on linear orderings, unpublished manuscript, 1980.

    Google Scholar 

  19. H. Herre, Linear Orderings in Stationary Logic, NTZ, Universitit Leipzig, Nr. 27 /91, 1991.

    Google Scholar 

  20. H. Herre, Theories of finitely determinate linear orderings in stationary logic, Quantifiers II M. Krynicki, M. Mostowski, L.W. Szczerba (eds.) Kluwer Academic Publ. 1995 pp. 89–113.

    Google Scholar 

  21. H. Herre and H. Wolter, Entscheidbarkeit der linearen Ordnung in L(Q1), Z. Math. Logik Grundl. Math. 23 (1977), pp. 273–282.

    Article  Google Scholar 

  22. H. Herre and H. Wolter, The Theory of Linear Orderings in Logics with Cardinality Quantifiers, P-06/80, P-13/80 (1980), IMath-AdW.

    Google Scholar 

  23. H. Herre and H. Wolter, Untersuchungen zur Theorie der linearen Ordnungen in Logiken mit Mâchtigkeitsquantoren, Zeitschr. f. Math. Logik and Grundl. d. Math. 27 (1981), pp. 73–94.

    Google Scholar 

  24. J. Hutchinson, Order types of ordinals in models of set theory, J. Symbolic Logic 41 (1976), pp. 489–502.

    Article  Google Scholar 

  25. T. Jech, Some combinatorial problems concerning uncountable cardinals , Ann. Math. Logic 5 (1973), pp. 165–198.

    Article  Google Scholar 

  26. M. Kaufmann, Some Results in Stationary Logic, Ph. D. Dissertation, University of Wisconsin, 1978.

    Google Scholar 

  27. H. J. Keisler, Logic with the quantifier “there exist uncountably many”, Annals of Math. Logic 1 (1970), pp. 1–93.

    Google Scholar 

  28. J. L. Krivine and K. Mcaloon, Forcing and generalized quantifiers, Annals of Mathematical Logic 5 (1973), pp. 199–255.

    Article  Google Scholar 

  29. D. W. Kueker, Countable approximations and L “wenhein-Skolem theorems, Ann. Math. Logic 11 (1977), pp. 57–104.

    Google Scholar 

  30. K. Kunen, Set Theory: An Introduction to Independency Proofs North-Holland Publ. Company, 1980, xvi + 313 pp.

    Google Scholar 

  31. C. K. Landraitis, Lω1ω equivalence between countable and uncountable linear orderings, FM 107 (1980), pp. 99–112.

    Google Scholar 

  32. H. Läuchli and J. Leonard, On the elementary theory of linear order, Fund. Math. 59 (1966), pp. 109–116.

    Google Scholar 

  33. P. Lindström, First order predicate logic with generalized quantifiers, Theoria 32 (1966), pp. 186–195.

    Google Scholar 

  34. M. Magidor and J.J. Malitz, Compact extensions of L(Q), Annals of Mathematical Logic 11 (1977), pp. 217–161.

    Article  Google Scholar 

  35. A. H. Mekler, Stationary logic of ordinals, Annals of pure and Applied Logic 26 (1984), pp. 47–68.

    Article  Google Scholar 

  36. A. Mostowski, On a generalization of quantifiers, Fund. Mathe. 44 (1957), pp. 12–36.

    Google Scholar 

  37. F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. (2) vol. 30 (1930), pp. 264–286.

    Article  Google Scholar 

  38. M. Rubin, Theory of linear order, Isreal Journal of Mathematics 17 (1974), pp. 392–443.

    Google Scholar 

  39. D. Seese, Stationary logic and ordinals,Trans. of the AMS 263 (1981),pp. 111–124.

    Google Scholar 

  40. D. G. Seese, H.-P. Tuschik and M. Weese, Undecid- able theories in stationary logic, Proc. of the ASM 84 (1982), pp. 563–567.

    Google Scholar 

  41. S. Shelah, The monadic theory of order, Annals of Mathematics 102 (1975), pp. 379–419.

    Google Scholar 

  42. S. Shelah, Generalized quantifiers and compact logic,Trans. Am.Math. Soc. 204 (1975) pp. 342–367.

    Google Scholar 

  43. S. Shelah, Categoricity in l j of sentences of L~ 1rW (Q), Israel J. Math. 20 (1975), pp. 127–148.

    Article  Google Scholar 

  44. S. Shelah, “L < “ is not countably compact” is consistent,Manuscript, Hebrew University.

    Google Scholar 

  45. A. B. Slomson, Generalized quantifiers and well-orderings, Arch. Math. Logik 15 (1972), pp. 57–73.

    Article  Google Scholar 

  46. S. Todorcevic Trees and linearly ordered sets, Handbook of Set-Theoretic Topology, K. Kunen, J.E. Vaughan (eds.), North-Holland, 1984, pp. 235–298.

    Google Scholar 

  47. P. H. Tuschik, On the Decidability of the Theory of Linear Orderings in the Language L(Q1), Lect. Notes in Math., vol. 619, Springer Verlag, Berlin-Heidelberg-New York, 1977, pp. 291–304.

    Google Scholar 

  48. H.-P. Tuschik, On the decidability of the theory of linear orderings with generalized quantifiers, FM 107 (1980), pp. 21–32.

    Google Scholar 

  49. H.-P. Tuschik, The expressive power of Malitz quantifiers for linear orderings,Annals of Mathematical Logic 36 (1987), pp. 53–103.

    Google Scholar 

  50. R. L. Vaught, The completeness of logic with added quantifier “there exist uncountably many”,Fund. Math. 54 (1964), 303 pp.

    Google Scholar 

  51. S. Vinner, A generalization of Ehrenfeucht’s game and some applications, Israel Journal of Math. 12 (1972) pp. 278–298.

    Google Scholar 

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Authors and Affiliations

  1. University of Leipzig, Germany

    Heinrich Herre

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  1. Heinrich Herre
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Warsaw, Poland

    Michał Krynicki

  2. Department of Philosophy, University of Warsaw, Poland

    Marcin Mostowski

  3. Siedlce University, Poland

    Lesław W. Szczerba

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Herre, H. (1995). Theory of Linear Order in Extended Logics. In: Krynicki, M., Mostowski, M., Szczerba, L.W. (eds) Quantifiers: Logics, Models and Computation. Synthese Library, vol 248. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0522-6_6

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  • DOI: https://doi.org/10.1007/978-94-017-0522-6_6

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4539-3

  • Online ISBN: 978-94-017-0522-6

  • eBook Packages: Springer Book Archive

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