Notre Dame Journal of Formal Logic

Bounds on Weak Scattering

Gerald E. Sacks

Abstract

The notion of a weakly scattered theory T is defined. T need not be scattered. For each $\cal A$ a model of T, let sr($\cal A$) be the Scott rank of $\cal A$. Assume sr($\cal A$) ≤ ω\sp A \sb 1 for all $\cal A$ a model of T. Let σ\sp T \sb 2 be the least Σ₂ admissible ordinal relative to T. If T admits effective k-splitting as defined in this paper, then $∃θ < σ\sp T \sb 2 such that sr($\cal A$) < θ for all $\cal A$ a model of T.

Article information

Source
Notre Dame J. Formal Logic, Volume 48, Number 1 (2007), 5-31.

Dates
First available in Project Euclid: 1 March 2007

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1172787542

Digital Object Identifier
doi:10.1305/ndjfl/1172787542

Mathematical Reviews number (MathSciNet)
MR2289894

Zentralblatt MATH identifier
1123.03021

Subjects
Primary: 03C70: Logic on admissible sets 03D60: Computability and recursion theory on ordinals, admissible sets, etc.

Keywords
weakly scattered theories bounds on Scott rank

Citation

Sacks, Gerald E. Bounds on Weak Scattering. Notre Dame J. Formal Logic 48 (2007), no. 1, 5--31. doi:10.1305/ndjfl/1172787542. https://projecteuclid.org/euclid.ndjfl/1172787542


Export citation

References

  • [1] Baldwin, J., "The Vaught Conjecture: Do Uncountable Models Count?", Notre Dame Journal of Formal Logic, vol. 48 (2007), pp. 79--92 (electronic).
  • [2] Barwise, J., Admissible Sets and Structures. An Approach to Definability Theory, Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1975.
  • [3] Chan, A., Models of High Rank for Weakly Scattered Theories, Ph.D. thesis, Massachusetts Institute of Technology, 2006.
  • [4] Goddard, C., Improving a Bounding Result for Weakly-Scattered Theories, Ph.D. thesis, Massachusetts Institute of Technology, 2006.
  • [5] Grilliot, T. J., "Omitting types: Application to recursion theory", The Journal of Symbolic Logic, vol. 37 (1972), pp. 81--89.
  • [6] Harnik, V., and M. Makkai, "A tree argument in infinitary model theory", Proceedings of the American Mathematical Society, vol. 67 (1977), pp. 309--14.
  • [7] Keisler, H. J., Model Theory for Infinitary Logic. Logic with Countable Conjunctions and Finite Quantifiers, vol. 62 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1971.
  • [8] Knight, J. F., and J. M. Young, "Computable structures of rank $\omega_1^\mathitCK$", preprint, 2004.
  • [9] Knight, R. W., "The Vaught Conjecture: A Counterexample", manuscript, 2002. http://www.maths.ox.ac.uk/~knight/stuff/example2.ps
  • [10] Makkai, M., "An `admissible' generalization of a theorem on countable $\Sigma \sp1\sb1$" sets of reals with applications, Annals of Pure and Applied Logic, vol. 11 (1977), pp. 1--30.
  • [11] Makkai, M., "An example concerning Scott heights", The Journal of Symbolic Logic, vol. 46 (1981), pp. 301--18.
  • [12] Millar, J., and G. E. Sacks, "Models with high Scott rank", in preparation.
  • [13] Morley, M., "The number of countable models", The Journal of Symbolic Logic, vol. 35 (1970), pp. 14--18.
  • [14] Nadel, M., "Scott sentences and admissible sets", Annals of Pure and Applied Logic, vol. 7 (1974), pp. 267--94.
  • [15] Sacks, G. E., "On the number of countable models", pp. 185--95 in Southeast Asian Conference on Logic (Singapore, 1981), vol. 111 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1983.
  • [16] Sacks, G. E., Higher Recursion Theory, Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1990.
  • [17] Scott, D., "Logic with denumerably long formulas and finite strings of quantifiers", pp. 329--41 in Theory of Models (Proceedings of the 1963 International Symposium, Berkeley), North-Holland, Amsterdam, 1965.
  • [18] Steel, J. R., "On Vaught's conjecture", pp. 193--208 in Cabal Seminar 76--77 (Proceedings of the Caltech-UCLA Logic Seminar, 1976--77), vol. 689 of Lecture Notes in Mathematics, Springer, Berlin, 1978.