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Xik_,} is the subtype of F in variables xio,... ,x~_,. , x,,_~) is complete in T if for every formula r x=_~), exactly one of T F (Wo)... , ~,_1 ) ~ r x,-, )], T ~ (Wo)... , x,,_,)] Chapter I Pure Computable Model Theory holds. That is, there is exactly one (complete) type of T in x 0 , . . , xn-1 which contains O. A type which contains a complete formula is called principal. A is called the type spectrum of J[. A type spectrum of a theory is the type spectrum of one of its models. Let X be a set.

Equivalently, ['(F) is computable if the set { n : 0,(F) E ['(~)} is computable. 2 Every type realized in a decidable model is computable. P r o o f . Let A be a decidable model such that a type F(x0,... A) is realized in A by some ao,... , an-1 C A. ,an-,), F must be computable, o A set of codes of a set of computable (complete) types of a theory T is a set of GSdel numbers of characteristic functions (which are computable) of these types, containing at least one index for each type. We say that a set of computable types belongs to P, where P is a complexity class, if it has a set of codes which belongs to P.

Clayton, Australia, Aug. 1-4, 1979), J. N. , Yarra Glen, Victoria, Australia, 1981) 147-160. [91] C. McCarty, Realizability and recursive set theory, Ann. Pure Appl. Logic, 32 (1986) 153-183. [92] A. I. Mal'tsev, Constructive algebras I (Russian), Uspekhi Mat. Nauk, 16 (1961) 3-60; [translated in: Constructive algebras I, Russian Math. Surveys, 16:3 (1961) 77-129; also in: The Metamathematics of Algebraic Systems, Collected Papers: 1936-1967, translated and edited by B. F. Wells llI, Stud. Logic Found.