Chapter 1 Pure computable model theory [chapter]

Valentina S. Harizanov
1998 Studies in Logic and the Foundations of Mathematics  
Shore; in Boolean algebras by Carroll, Feiner, LaRoche, Remmel and Thurber; in topological spaces by Kalantari, Legett, Remmel, Retzlaff and Weitkamp. Computable Ramsey's theory has been studied by Clote, Hummel, Jockusch, Seetapun, Simpson, Solovay and Specker. Computability in analysis and physics has also been studied, see [176] . The generalization of the definition of a particular computable algebraic structure to an arbitrary model yields one of the basic concepts of pure computable model
more » ... theory, an area of logic developed in the last twenty-five years. That is, the notion of a computable model, and a stronger notion of a decidable model. Lerman and Schmerl have given examples of theories with computable models. The first general results in computable model theory have been obtained by following the fundamental notions and results of classical model theory. For example, Millar has obtained the effective version of the omitting types theorem, and Harrington, Goncharov and Nurtazin have found when a complete decidable theory with a prime model has a decidable prime model. Millar and Morley have characterized decidable theories with decidable saturated models, and Goncharov and Peretyat'kin have characterized decidable homogeneous models. Barwise, Schlipf and Ressayre have introduced the notion of a computably saturated model. Although developed in the context of admissible sets and admissible fragments of infinitary logic, computably saturated models have also provided a useful tool for research and exposition in classical model theory. In the West, Millar has further produced an extensive body of work on topics including effective Vaught's theorem, the structure of types in decidable models, decidability and prime, saturated and homogeneous models, decidable theories with finitely many and decidable theories with countably many non-isomorphic countable models. Reed has also studied decidable theories with finitely many non-isomorphic countable models. Kierstead and Remmel have investigated the degrees of sets of indiscernibles in decidable models. Ash, Knight, Macintyre, Marker, Nadel, Nies, Richter, Jockusch, Lachlan, Scott, Shoenfield, Shore, Soare and Tennenbaum have studied the degrees of models of various theories, including the theory of linear orders, Peano arithmetic, true arithmetic, and the theory of Boolean algebras. The whole spectrum of questions involving the isomorphisms of abstract computable models has been investigated The lattices of computably enumerable submodels have been studied by Ash, Guichard, Carroll, Downey, Metakides, Nerode, Remmel and Smith. More recently, Nerode, Remmel and Cenzer [31, 158] have been developing feasible model theory (as a part of feasible mathematics), the theory of models with bounded space and time resources. They have investigated how feasible models differ from computable models. The feasible models studied include Boolean algebras, abelian groups, linear orders, models of arithmetic, and graphs. , φ (n),X 2 , . . . be a fixed effective enumeration of all n-ary X-computable functions. The superscripts are usually omitted for n = 1 or when it is clear from the context. φ e (φ X e ) is also denoted by {e} ({e} X ), and e is called the Gödel number or index of φ e . We write φ e,s (n) = m if e, n, m < s and m is the output of φ e (n) after < s steps in the corresponding computation. Let W e = def dom(φ e ) and W e,s = def dom(φ e,s ). Thus, W 0 , W 1 , W 2 , . . . is a computable enumeration of all c.e. sets. We fix h · , · i to be a computable bijection from ω 2 onto ω, which is strictly increasing with respect to both arguments. For X ⊆ ω and i ∈ ω, we define We assume that a formula is identified with its Gödel number, so a set of formulae is thought of as a subset of ω. Thus, a theory is decidable (resp. belongs to P, where P is a complexity class) if the set of Gödel numbers of its sentences is computable (resp. belongs to P). Hence, if Ax is a set of axioms of a theory T , then T is decidable if there is an algorithm which determines for every sentence σ of L, whether Ax'σ. Clearly, a computably axiomatizable theory is computably enumerable. Hence a complete computably axiomatizable theory is decidable. In particular, a complete finitely axiomatizable theory is decidable. An example of such a theory is the theory of dense linear order. Peretyat'kin [168, 169, 170, 171, 172, 173, 174] has developed intricate methods for constructing finitely axiomatizable theories satisfying various additional properties. In [168] , he constructed a complete, finitely axiomatizable, ℵ 1 -categorical theory which is not ℵ 0 -categorical. Well-known and important examples of decidable theories in mathematics include the theory of equality, the theory of unary predicates, the additive number theory, the theory of the field of real numbers, the theory of the field of complex numbers, the theory of algebraically closed fields, the theory of real-closed fields, the theory of p-adic fields, the theory of Boolean algebras, the theory of linear order, the theory of abelian groups, and the theory of free commutative algebras. Well-known and important examples of undecidable theories in mathematics include number theory, the theory of simple groups, the theory of semigroups, the theory of rings, the theory of fields, the theory of distributive lattices, and the theory of partial order. For more information on decidable and undecidable theories see [58] and Part III in [150] . For computability theoretic complexity of various sets of sentences satisfied in certain classes of models see [204] . A model A is computable if A is computable, and if there is a computable enumeration (a i ) i∈ω of A and an algorithm which determines, for every quantifierfree formula θ(x 0 , . . . , x n−1 ) and every sequence (a i0 , . . . , a in−1 ) ∈ A n , whether A A |= θ(a i 0 , . . . , a i n−1 ). A model A is decidable if A is computable and there is a computable enumeration (a i ) i∈ω of A and an algorithm which determines for every formula θ(x 0 , . . . , x n−1 ) and every sequence (a i0 , . . . , a in−1 ) ∈ A n , whether A A |= θ(a i0 , . . . , a in−1 ). Clearly, every decidable model is computable. The converse is not true. For example, (ω, +, ×) is a computable model which is not decidable (by Gödel's incompleteness theorem [64] ). Peretyat'kin [163] has constructed a decidable linear order without a computable proper elementary extension. In [160], Nurtazin characterized decidable models which are isomorphic to computable non-decidable models. Peretyat'kin [166] has shown that there is a complete decidable theory T which is neither ℵ 0 -categorical nor ℵ 1 -categorical, and which has, up to isomorphism, a unique decidable model. Moreover, all computable models of T are decidable. A model is computably presentable if it is isomorphic to a computable model. Goncharov [66] has constructed an ℵ 1 -categorical theory which is not ℵ 0categorical and whose only computably presentable model is the prime model. On the other hand, Khoussainov, Nies and Shore [101] have shown that there is an ℵ 1 -categorical theory which is not ℵ 0 -categorical and whose only countable non-computably presentable model is the prime model. It is sometimes conve-Proposition 4.7. The set of all types of T realized in a decidable model of T is computable. Proof. Let A be a decidable model of T and let a 0 , a 1 , a 2 , . . . be an effective enumeration of A. Choose g : A <ω → ω to be a computable bijection. Define a computable function h : ω 2 → {0, 1} by:
doi:10.1016/s0049-237x(98)80002-5 fatcat:mktikwmmazgale5ejrpnihv6hm