Notre Dame Journal of Formal Logic

Improving a Bounding Result That Constructs Models of High Scott Rank

Christina Goddard

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1445606158

Digital Object Identifier
doi:10.1215/00294527-3328289

Mathematical Reviews number (MathSciNet)
MR3447725

Zentralblatt MATH identifier
06550120

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

Keywords
raw tree hierarchy weakly scattered theories bounds on Scott rank

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


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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.