Computable Content of Vaughtian Models

Vaught's models [1961]

In what David Marker [2002, p. 172] calls "one of the most elegant papers in model theory," Robert Vaught [1961] developed the theory of prime, saturated, and homogeneous models using types. These models, which we call Vaughtian models , have played an important role in model theory.

During the last few years there have been a number of results about the computable content of these models. The techniques from both model theory and computability are particularly elegant and not difficult. For maximum benefit the students should master some elementary and basic material before the lectures, although we shall review them briefly at the start of each lecture.

Conference on Vaught's Conjecture

Julia Knight hosted a conference in May, 2005 at Notre Dame on Vaught's Conjecture and Peter Cholak is editing a volume of the papers presented. Karen Lange nd Robert Soare have written a paper Computability of Homogeneous Models with both the background results and new results mostly by Karen Lange on the computable content of homogeneous models. This paper will appear in the Notre DAme Journal of Formal Logic.

Tutorial on Computable Model Theory

Robert Soare prepared a two chapter tutorial on model theory and computability.

Tutorial on Vaughtian Models and computability results for graduate students in mathematical logic. No previous background is necessary.

Chapter 1 on Vaught's Models

Chapter 2 on Computability Theory necessary for computable model theory.

Lecture on Homogeneous and Saturated Models

Robert Soare gave a twenty minute lecture at the AMS meeting in honor of Manuel Lerman, UConn, Storrs CT, October 29, 2006.

ABSTRACT: Degrees of Homogeneous Models

Vaught [1961] defined a model to be homogeneous if every finite partial elementary map can be extended to an automorphism. Goncharov and Peretyatkin found a criterion for a homogeneous model with all types uniformly effectively presented to have a decidable copy. A number of results by researchers at the University of Chicago considerably improve these results in the positive and negative direction. We shall describe some of them. Most are due to University of Chicago graduate students, Karen Lange and Ken Harris.

Lectures on Computable Content of Vaughtian Models

Robert Soare gave four lectures during August 21-25, 2006 at Leeds on the topic of computable content of Vaughtian models. Attached here are the transparencies of his slides in .pdf form as a guide for the lectures but the actual lectures will contain many more stories, diagrams, and information not on the slides.

Lecture 1: Degrees of Prime Models

Lecture 2: Degrees Bounding Prime Models

Lecture 3: Degrees of Saturated Models

Lecture 4: Degrees of Homogeneous Models

References

[Marker, 2002]
D.~Marker, Model Theory: An Introduction, Graduate Texts in Mathematics, Springer-Verlag, New York, 2002.

[Soare, 1987]
R.~I.~Soare, Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets, Springer-Verlag, Heidelberg, 1987.

[Soare, cta]
R.~I.~Soare, Computability Theory and Applications Springer-Verlag, Heidelberg, under contract with Springer, in preparation.

[Vaught, 1961]
R.~L.~Vaught, Denumerable models of complete theories, Proceedings of Symposium on Foundations ofMathematics: Infinistic Methods , Pergamon Press, London, 301--321.