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  • Cited by 11
  • Print publication year: 1999
  • Online publication date: May 2010

Effective Model Theory: The Number of Models and Their Complexity

Summary

Abstract

Effective model theory studies model theoretic notions with an eye towards issues of computability and effectiveness. We consider two possible starting points. If the basic objects are taken to be theories, then the appropriate effective version investigates decidable theories (the set of theorems is computable) and decidable structures (ones with decidable theories). If the objects of initial interest are typical mathematical structures, then the starting point is computable structures. We present an introduction to both of these aspects of effective model theory organized roughly around the themes of the number and types of models of theories with particular attention to categoricity (as either a hypothesis or a conclusion) and the analysis of various computability issues in families of models.

Basic Notions

The lectures on which this paper is based were intended to be a brief introduction to effective model theory centered around one set of issues: the number of models of specified type and, in particular, the notion of categoricity. For more general introductions we refer the reader to The Handbook of Recursive Algebra (Ershov et al. [1998]), especially the articles by Harizanov [1998] and Ershov and Goncharov [1998]. This Handbook also contains other useful survey papers on aspects of effective model theory and algebra and an extensive bibliography. The one most closely related to the theme of this paper is Goncharov [1998]. Another interesting survey is Millar [1999] in The Handbook of Computability Theory (Griffor [1999]).