## Notre Dame Journal of Formal Logic

### Improving a Bounding Result That Constructs Models of High Scott Rank

Christina Goddard

#### Abstract

Let $T$ be a theory in a countable fragment of $\mathcal{L}_{\omega_{1},\omega}$ 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 $T_{\gamma}\subseteq T_{\delta}$ with $T_{\gamma}$- and $T_{\delta}$-theories on level $\gamma$ and $\delta$, respectively. Then if $\ulcorner p\urcorner (T_{\delta})$ is a formula that defines a type for $T_{\delta}$, our predecessor function provides a formula for defining its subtype in $T_{\gamma}$.

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

#### Article information

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

Dates
Accepted: 25 September 2013
First available in Project Euclid: 23 October 2015

https://projecteuclid.org/euclid.ndjfl/1445606158

Digital Object Identifier
doi:10.1215/00294527-3328289

Mathematical Reviews number (MathSciNet)
MR3447725

Zentralblatt MATH identifier
06550120

#### Citation

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. https://projecteuclid.org/euclid.ndjfl/1445606158

#### References

• [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.