Notre Dame Journal of Formal Logic

Improving a Bounding Result That Constructs Models of High Scott Rank

Christina Goddard

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Let T be a theory in a countable fragment of Lω1,ω whose extensions in countable fragments have only countably many types. Sacks proves a bounding theorem that generates models of high Scott rank. For this theorem, a tree hierarchy is developed for T that enumerates these extensions.

In this paper, we effectively construct a predecessor function for formulas defining types in this tree hierarchy as follows. Let TγTδ with Tγ- and Tδ-theories on level γ and δ, respectively. Then if p(Tδ) is a formula that defines a type for Tδ, our predecessor function provides a formula for defining its subtype in Tγ.

By constructing this predecessor function, we weaken an assumption for Sacks’s result.

Article information

Notre Dame J. Formal Logic, Volume 57, Number 1 (2016), 59-71.

Received: 17 October 2011
Accepted: 25 September 2013
First available in Project Euclid: 23 October 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

raw tree hierarchy weakly scattered theories bounds on Scott rank


Goddard, Christina. Improving a Bounding Result That Constructs Models of High Scott Rank. Notre Dame J. Formal Logic 57 (2016), no. 1, 59--71. doi:10.1215/00294527-3328289.

Export citation


  • [1] Barwise, J., Admissible Sets and Structures: An Approach to Definability Theory, vol. 7 of Perspectives in Mathematical Logic, Springer, Berlin, 1975.
  • [2] Goddard, C. M., “Improving a bounding result for weakly-scattered theories,” Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, Mass., 2006.
  • [3] Morley, M., “The number of countable models,” Journal of Symbolic Logic, vol. 35 (1970), pp. 14–18.
  • [4] Sacks, G. E., “Bounds on weak scattering,” Notre Dame Journal of Formal Logic, vol. 48 (2007), pp. 5–31.