# Elementary theory

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A collection of closed formulas of first-order predicate logic. The elementary theory of a class of algebraic systems (cf. Algebraic system) of signature is defined to be the collection of all closed formulas of the first-order predicate logic of signature that are true in all systems of . If consists of a single system , then the elementary theory of the class is the elementary theory of the system . Two algebraic systems of the same signature are said to be elementarily equivalent if their elementary theories are the same. An algebraic system of signature is called a model of an elementary theory of signature if all formulas of are true in . An elementary theory is called consistent if it has models. A consistent elementary theory is called complete if any two models of it are elementarily equivalent. The class of all models of an elementary theory is denoted by . An elementary theory is called solvable (or decidable) if the set of formulas (that is, the set of all logical consequences of ) is recursive. A class of algebraic systems of signature is called axiomatizable if there exists an elementary theory of signature such that . In this case is called a collection of axioms for . A class is axiomatizable if and only if . For example, the class of dense linear orders without a smallest or largest element is axiomatizable, its elementary theory is solvable and any two systems of this class are elementarily equivalent, since the elementary theory of this class is complete; moreover, its elementary theory is finitely axiomatizable. The class of finite cyclic groups is not axiomatizable; however, its elementary theory is solvable, and hence recursively axiomatizable. There are examples of finitely-axiomatizable unsolvable elementary theories. They include those of groups, rings, fields, and others. However, a complete recursively-axiomatizable elementary theory is necessarily solvable. Therefore, to prove the solvability of a recursively-axiomatizable elementary theory it is sufficient to observe that it is complete.
Several methods for proving completeness are known. An elementary theory is called categorical in cardinality (cf. Categoricity in cardinality) if all its models of cardinality are isomorphic. An elementary theory that is categorical in some infinite cardinality and has no finite models is necessarily complete. For example, the elementary theory of algebraically closed fields of a given characteristic is recursively axiomatizable and categorical in every uncountable cardinality; it has no finite models, and therefore it is complete and solvable. In particular, the elementary theory of the field of complex numbers is solvable. Two formulas in the same signature as that of a theory are equivalent in the theory if they contain the same variables and if, for any model of and any assignment of elements of to their free variables, the formulas are either both true or both false. A complete elementary theory of finite or countable signature is countably categorical if and only if for every there are finitely many formulas with free variables such that every formula of the appropriate signature with as free variables is equivalent in to one of those formulas. A complete theory of finite or countable signature that is categorical in one uncountable cardinality is also categorical in every other uncountable cardinality. A system of signature is called an elementary subsystem of a system of the same signature if is a subsystem of and if for every formula of the first-order predicate logic of with free variables and all , the truth of in implies its truth in . An elementary theory is called model complete if for any two models and of it the fact that is a subsystem of implies that it is an elementary subsystem. It turns out that a model-complete theory having a model that can be isomorphically imbedded in every model of the theory is complete. Two systems of the same signature which satisfy the same prenex formulas without existential quantifiers are called universally equivalent. A model-complete elementary theory all models of which are universally equivalent is complete. Using the technique of model completeness one can prove that real-closed fields, in particular, the field of real numbers, have a complete and solvable elementary theory. Among the other solvable elementary theories are those of addition of natural numbers and integers, of Abelian groups, of -adic number fields, of finite fields, of residue class fields, of ordered Abelian groups, and of Boolean algebras.
The general study of unsolvable elementary theories was initiated by A. Tarski in the 1940s, but even earlier, in 1936, A. Church had proved the unsolvability of first-order predicate logic and J. Rosser, also in 1936, had proved the unsolvability of the arithmetic of the natural numbers. The elementary theory of a class of algebraic systems of the same signature is said to be inseparable if there is no recursive set of formulas containing and not containing any closed formula which is false in all systems in . The elementary theory of a class of systems of signature consisting of a single two-place predicate is called relatively definable in the elementary theory of a class of systems of signature if there exist formulas and of such that for every system in one can find a system in and elements in for which the set together with the predicate , defined on so that is true if and only if is true in , forms an algebraic system isomorphic to . This definition extends naturally to the theory of classes of arbitrary signature. If the elementary theory of a class is inseparable and relatively definable in the elementary theory of a class , then that of is also inseparable. This makes it possible to prove that the elementary theories of many classes of algebraic systems are inseparable. Here it is convenient to take as the elementary theory of all finite binary relations or that of all finite symmetric relations, or similar elementary theories. Inseparable elementary theories are unsolvable. So are those of the field of rational numbers and of many classes of rings and fields. The unsolvability of the elementary theory of finite groups is an important result of A.I. Mal'tsev.