## Notre Dame Journal of Formal Logic

### Bounds on Weak Scattering

Gerald E. Sacks

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$a model of T. #### Article information Source Notre Dame J. Formal Logic, Volume 48, Number 1 (2007), 5-31. Dates First available in Project Euclid: 1 March 2007 Permanent link to this document https://projecteuclid.org/euclid.ndjfl/1172787542 Digital Object Identifier doi:10.1305/ndjfl/1172787542 Mathematical Reviews number (MathSciNet) MR2289894 Zentralblatt MATH identifier 1123.03021 #### Citation Sacks, Gerald E. Bounds on Weak Scattering. Notre Dame J. Formal Logic 48 (2007), no. 1, 5--31. doi:10.1305/ndjfl/1172787542. https://projecteuclid.org/euclid.ndjfl/1172787542 #### References • [1] Baldwin, J., "The Vaught Conjecture: Do Uncountable Models Count?", Notre Dame Journal of Formal Logic, vol. 48 (2007), pp. 79--92 (electronic). • [2] Barwise, J., Admissible Sets and Structures. An Approach to Definability Theory, Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1975. • [3] Chan, A., Models of High Rank for Weakly Scattered Theories, Ph.D. thesis, Massachusetts Institute of Technology, 2006. • [4] Goddard, C., Improving a Bounding Result for Weakly-Scattered Theories, Ph.D. thesis, Massachusetts Institute of Technology, 2006. • [5] Grilliot, T. J., "Omitting types: Application to recursion theory", The Journal of Symbolic Logic, vol. 37 (1972), pp. 81--89. • [6] Harnik, V., and M. Makkai, "A tree argument in infinitary model theory", Proceedings of the American Mathematical Society, vol. 67 (1977), pp. 309--14. • [7] Keisler, H. J., Model Theory for Infinitary Logic. Logic with Countable Conjunctions and Finite Quantifiers, vol. 62 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1971. • [8] Knight, J. F., and J. M. Young, "Computable structures of rank$\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.
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