Chapter 6 Autostable models and algorithmic dimensions

Abstract

Publisher Summary This chapter discusses autostable models and algorithmic dimensions. Any constructivizable structure whose group of automorphisms has the cardinality of the continuum has many nonrecursively equivalent constructivizations. Thus, despite the simple algorithmic structure of a countable atomless Boolean algebra, many continuum nonrecursively equivalent constructivizations are obtained. For most classes of models and algebras, the algorithmic dimension is infinite. They are of a model-complete character and in a number of natural classes of models and algebras allow giving complete characterization of algebraic structures of infinite algorithmic dimension, while their negations characterize autostable structures. The first criterion arose in the study of algorithmic dimensions of torsion-free Abelian groups and vector spaces. Its informal idea is to find an infinite number of limit points in the structure with respect to the topology defined by quantifier-free formulas. The chapter provides a recursion-theoretic criterion for an algorithmic dimension to be infinite. This criterion is of great interest because it provides a new point of view on the question of the number of nonautoequivalent constructivizations.