Non-computable Models of Certain First Order Theories Gábor Sági Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13-15, H-1053 Budapest, Hungary and Budapest University of Technology and Economics, Department of Algebra, Egry J. u. 1, H-1111 Budapest, Hungary Email: sagi@renyi.hu Ramón Horváth Instanridge AB, 111 45 Stockholm, Sweden, Birger Jarlsgatan 43, Email: haromn@gmail.com Abstract— Let D be a complexity class. A countable first order structure is defined to be D-presented iff all of its basic relations and functions are in D. We show, that if T is a first order theory with at least one uncountable Stone space then T has a countable model not isomorphic to any D-presented one. We also show that there is a countable ℵ0-categorical structure in a finite language which is not isomorphic to any D-presented structure; in addition, there exists a consistent first order theory in a finite language that does not have D-presented models, at all. Our proofs utilize model theoretic methods and do not involve any nontrivial recursion theoretic notion or construction. AMS Subject Classification: 03C57, 03D45. Keywords: Computable structures, complexity classes, ℵ0-categorical structures, oligomorphic permutation groups. I. I NTRODUCTION Abstract data types (for example, in object oriented programming languages) may be regarded as certain first order structures with countable universe and a set of computable operations and relations in it. Further, very often it is important to implement algorithms capable to perform op- erations in certain countable first order structures (such as algebraic number fields, rings, countably infinite Boolean Algebras, or groups, etc.). Imple- menting such algorithms have theoretical limita- tions: all the operations and relations should be recursive in the algorithm theoretical sense. In this paper we investigate such theoretical limitations, the main results are Theorems 4.1 and 5.13; they will be recalled below in the Intorduction after some technical preparations. A first order structure A is defined to be com- putable iff its universe is the set of natural numbers and all of its basic relations and functions are recursive (in the computational theoretic sense). More generally, if D is a complexity class (like the set of recursive, or the set of recursively enumerable relations), then a countable structure A is defined to be D-presented iff all of its basic relations and functions belong to D. For a more precise definition we refer to Section II below. By some classical results of Ershov, Arslanov and others, there are countable orderings, Boolean algebras, etc. which are not isomorphic to any computable structure. One of the main aims of model theory is to describe all structures in which a given theory (i.e. set of first order formulas) is true. At that level of generality this ambitious aim seems to be untractable. Hence, instead of it, model theorists are trying to characterize those theories which have a structure theorem, that is, whose models can be described in a comprehensive way. Recently, related investigations are very active. Along the results of Morley, Shelah, Lascar, Hrushovski, Cherlin, Pillay and others, it turned out, that theories have a “structure theoretic” hierarchy of complexity: in some cases the possible models are relatively easy to describe, in some other cases this is much more difficult, while in some other cases such a complete “comprehensive” description of