A Bird’s-Eye View of Twilight Combinatorics

  • J. C. E. Dekker
Part of the Progress in Computer Science and Applied Logic book series (PCS, volume 12)


Ordinary combinatorics deals with finite sets of numbers (nonnegative integers) and with finite collections of objects which can be coded by numbers. Many principles of ordinary combinatorics, e.g., the sum rule, the product rule, the inclusion-exclusion principle and the pigeonhole principle, have natural generalizations to isolated sets. We shall examine some of these generalizations and illustrate their use.


Product Rule Finite Subset Recursive Function Pigeonhole Principle Friendship Relation 
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  1. 1.
    Applebaum, C.H. and J.C.E. Dekker [1970], Partial recursive functions and ω-functions. J. Symbolic Logic 35, 559–568.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Barback, J. [1986], Tame models in the isols. Houston J. Math. 12, 163–175. For a correction see the same journal, 13, 301–302 (1987).MathSciNetzbMATHGoogle Scholar
  3. 3.
    Crossley, J.N. and A. Nerode [1974], Combinatorial functors. Springer-Verlag, New York.zbMATHCrossRefGoogle Scholar
  4. 4.
    Davis, M. [1958], Computability and unsolvability. McGraw-Hill, New York.zbMATHGoogle Scholar
  5. 5.
    Dekker, J.C.E. and Myhill, J. [1960], Recursive equivalence types. U. of California Publications in Math (N.S.), 3, 67–214.MathSciNetGoogle Scholar
  6. 6.
    Dekker, J.C.E. [1966], Les fonctions combinatoires et les isols. Gauthier Villars, Paris.zbMATHGoogle Scholar
  7. 7.
    Dekker, J.C.E. [1967], Regressive isols. In: Sets, models and recursion theory (J.N. Crossley, ed.), 272–296, North-Holland, Amsterdam.CrossRefGoogle Scholar
  8. 8.
    Dekker, J.C.E. [1981], Twilight graphs. J. Symbolic Logic 46, 539–571.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Dekker, J.C.E. [1986], The inclusion-exclusion principle for finitely many isolated sets. J. Symbolic Logic 51, 435–447.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Dekker. J.C.E. and E. Ellentuck [1989], Isols and the pigeonhole principle. J. Symbolic Logic 54, 833–846.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Hay, L.S., The cosimple isols [1966]. Annals of Mathematics 83, 231–256.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Hay, L.S. [1967], Elementary differences between the isols and the cosimple isols. Transactions of the AMS, 127, 427–441.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Heck, W.S. [1983], Large families of incomparable A-isols. J. Symbolic Logic 48, 250–252.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    McLaughlin, T.G. [1982], Regressive sets and the theory of isols. Marcel Dekker, New York.zbMATHGoogle Scholar
  15. 15.
    McLaughlin, T.G. [1987], Embeddings of and into Nerode semi-rings. Israel J. of Math. 60, 65–88.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Müller, G.H. and W. Lenski [1987], Ω-Bibliography of mathematical logic, 4 (P.G. Hinman, ed.), Springer, New York.Google Scholar
  17. 17.
    Myhill, J. [1958, 1962], Recursive equivalence types and combinatorial functions, Part 1, Bull, of the AMS 64, 373–376. Part 2, Logic, methodology and philosophy of science, Proc. of the 1960 Intl. Congress (E. Nagel, P. Suppes, A. Tarski, eds.), Stanford University Press, 46–55.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Nerode, A. [1961], Extensions to isols. Annals of Mathematics, 73, 362–403.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Nerode, A. [1962], Arithmetically isolated sets and nonstandard models, in Recursive function theory, Proc. of Symposia in Pure Mathematics, 5, AMS, Providence, 105–116.CrossRefGoogle Scholar
  20. 20.
    Nerode, A. [1966], Diophantine correct non-standard models in the isols. Annals of Mathematics 84, 421–432.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Nerode, A. and J.B. Remmel [1990], Polynomial time equivalence types, in Logic and Computation (W. Siegel, ed.), Contemporary Mathematics 706, AMS, Providence, 221–249.CrossRefGoogle Scholar
  22. 22.
    Nerode, A. and J.B. Remmel [1990], Polynomially isolated sets, in Recursion Theory Week (K. Ambos-Spies, G. Müller, G.E. Sacks, eds.). Lecture Notes in Mathematics 1432, Springer, New York, 323–362.CrossRefGoogle Scholar
  23. 23.
    Odifreddi, P. [1989], Classical recursion theory. North-Holland, Amsterdam.zbMATHGoogle Scholar
  24. 24.
    Post, E.L. [1944], Recursively enumerable sets of positive integers and their decision problems. Bulletin of the AMS, 50, 284–316.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Remmel. J.B. [1976], Combinatorial functors on co-r.e. structures. Annals of Math. Logic, 10, 261–287.MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Rogers, Jr., H. [1967], Theory of recursive functions and effective computability. McGraw-Hill, New York; 2nd edn., MIT Press, Cambridge (1987).zbMATHGoogle Scholar
  27. 27.
    Soare, R.I. [1987], Recursively enumerable sets and degrees. Springer, New York.Google Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • J. C. E. Dekker
    • 1
    • 2
  1. 1.Institute for Advanced StudyPrincetonUSA
  2. 2.Department of MathematicsRutgers UniversityNew BrunswickUSA

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