Skip to main content Accessibility help

Computability of Fraïssé limits

  • Barbara F. Csima (a1), Valentina S. Harizanov (a2), Russell Miller (a3) (a4) and Antonio Montalbán (a5)


Fraïssé studied countable structures through analysis of the age of , i.e., the set of all finitely generated substructures of . We investigate the effectiveness of his analysis, considering effectively presented lists of finitely generated structures and asking when such a list is the age of a computable structure. We focus particularly on the Fraïssé limit. We also show that degree spectra of relations on a sufficiently nice Fraïssé limit are always upward closed unless the relation is definable by a quantifier-free formula. We give some sufficient or necessary conditions for a Fraïssé limit to be spectrally universal. As an application, we prove that the computable atomless Boolean algebra is spectrally universal.



Hide All
[1]Ash, C. J. and Knight, J. F., Computable structures and the hyperarithmetical hierarchy, Elsevier, Amsterdam, 2000.
[2]Ash, C. J. and Nerode, A., Intrinsically recursive relations, Aspects of effective algebra (Crossley, J. N., editor), Upside Down A Book Company, Steel's Creek, Australia, 1981, pp. 2641.
[3]Chisholm, J., The complexity of intrinsically r.e. subsets of existentially decidable models, this Journal, vol. 55 (1990), pp. 12131232.
[4]Downey, R. G., Goncharov, S. S., and Hirschfeldt, D. R., Degree spectra of relations on boolean algebras, Algebra and Logic, vol. 42 (2003), pp. 105111.
[5]Downey, R. G. and Jockusch, C. G. Jr., Every low Boolean algebra is isomorphic to a recursive one, Proceedings of the American Mathematical Society, vol. 122 (1994), pp. 871880.
[6]Downey, R. G. and Knight, J. F., Orderings with α-th jump degree 0α, Proceedings of the American Mathematical Society, vol. 114 (1992), pp. 545552.
[7]Fraïssé, R., Theory of relations, North-Holland, Amsterdam, New York, 1986.
[8]Goncharov, S. S., Harizanov, V. S., Knight, J. F., McCoy, C., Miller, R. G., and Solomon, R., Enumerations in computable structure theory, Annals of Pure and Applied Logic, vol. 136 (2005), pp. 219246.
[9]Harizanov, V. S., Uncountable degree spectra, Annals of Pure and Applied Logic, vol. 54 (1991), pp. 255263.
[10]Harizanov, V. S., Pure computable model theory, Handbook of recursive mathematics (Ershov, Yu.L., Goncharov, S. S., Nerode, A., and Remmel, J. B., editors), vol. 1, Elsevier, Amsterdam, 1998, pp. 3114.
[11]Harizanov, V. S., Relations on computable structures, Contemporary mathematics, University of Belgrade, 2000, pp. 6581.
[12]Harizanov, V. S. and Miller, R. G., Spectra of structures and relations, this Journal, vol. 72 (2007), pp. 324348.
[13]Hirschfeldt, D. R., Khoussainov, B., Shore, R. A., and Slinko, A. M., Degree spectra and computable dimensions in algebraic structures, Annals of Pure and Applied Logic, vol. 115 (2002), pp. 71113.
[14]Hodges, W., A shorter model theory, Cambridge University Press, Cambridge, 1997.
[15]Jockusch, C. G. Jr. and Soare, R. I., Degrees of orderings not isomorphic to recursive linear orderings, Annals of Pure and Applied Logic, vol. 52 (1991), pp. 3964.
[16]Khoussainov, B. and Shore, R. A., Computable isomorphisms, degree spectra of relations, and Scott families, Annals of Pure and Applied Logic, vol. 93 (1998), pp. 153193.
[17]Khoussainov, B. and Shore, R. A., Effective model theory: The number of models and their complexity, Models and computability: Invited papers from Logic Colloquium '97 (Cooper, S. B. and Truss, J. K., editors), London Mathematical Society Lecture Notes Series, vol. 259, Cambridge University Press, Cambridge, 1999, pp. 193240.
[18]Knight, J. F., Degrees coded in jumps of orderings, this Journal, vol. 51 (1986), pp. 10341042.
[19]Miller, R. G., The Δ20-spectrum of a linear order, this Journal, vol. 66 (2001), pp. 470486.
[20]Moses, M. F., Relations intrinsically recursive in linear orders, Zeitschrift für Mathematische Logik und Grundtagen der Mathematik, vol. 32 (1986), pp. 467472.
[21]Richter, L. J., Degrees of structures, this Journal, vol. 46 (1981), pp. 723731.
[22]Slaman, T., Relative to any nonrecursive set, Proceedings of the American Mathematical Society, vol. 126 (1998), pp. 21172122.
[23]Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, New York, 1987.
[24]van der Waerden, B. L., Moderne Algebra, Springer, Berlin, 1930, English translation Algebra, (F. Blum and J. R. Schulenberger, translators) Springer-Verlag, New York, 1991.
[25]Wehner, S., Enumerations, countable structures, and Turing degrees, Proceedings of the American Mathematical Society, vol. 126 (1998), pp. 21312139.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed