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We introduce a tool for analysing models of $\text {CT}^-$ , the compositional truth theory over Peano Arithmetic. We present a new proof of Lachlan’s theorem that the arithmetical part of models of $\text {CT}^-$ are recursively saturated. We also use this tool to provide a new proof of theorem from [8] that all models of $\text {CT}^-$ carry a partial inductive truth predicate. Finally, we construct a partial truth predicate defined for a set of formulae whose syntactic depth forms a nonstandard cut which cannot be extended to a full truth predicate satisfying $\text {CT}^-$ .

The paper presents an outline of the general theory of countable arithmetically saturated models of PA and some of its applications. We consider questions concerning the automorphism group of a countable recursively saturated model of PA. We prove new results concerning fixed point sets, open subgroups, and the cofinality of the automorphism group. We also prove that the standard system of a countable arithmetically saturated model of PA is determined by the lattice of its elementary substructures.is Th(N). The standard system of a model M , SSy(M ), is the family of those X ⊆ M for which there is an Y definable in M with parameters, such that X = ω ∩ Y. L PA will denote the language of PA.We will say that a type p( v, ā) in variables v = v 1 , . . . , v n , and parameters ā = a 1 , . . . , a m ∈ M is recursive, arithmetic, etc., if the set of Gödel numbers of formulas ϕ( v, w) ∈ p( v, w), where w = w 1 , . . . , w m , is recursive, arithmetic, etc. In the same sense we will speak of types as being subsets of ω.The notion of A-saturation was introduced by Wilmers in [16]. Let A be a family of subsets of ω. We say that a model M is A-saturated if the following two conditions are satisfied: (i) for every ā ∈ [M ] <ω , the type of ā, tp( ā), is in A; (ii) for every type p( v, ā) in A, if p( v, ā) is realized in some elementary extension of M , then it is realized in M .A Scott set is an ω-model of WKL 0 . The standard system of a model of PA is a Scott set; moreover, every countable Scott set is the standard system of a model of PA. If T is a completion of PA, X is a countable Scott set and T ∈ X , then there is a recursively saturated countable model M |= T such that SSy(M ) = X . Proofs of the above statements can be found in Kaye [2].The next proposition shows that a recursively saturated model of PA is much more than just recursively saturated.We will use a fixed arithmetical coding of finite sequences. If M is a model of PA and a, i ∈ M , then (a) i denotes the i-th term of the sequence coded by a, and lena is the length of the sequence coded by a.

Abstract.Using a weak notion of recursive saturation (not always semiregularity) we prove that there are no finitely generated countable models of B~Ln + ->/!"(« > 0). We consider the problem of not almost semiregularity of models of /X" + -"5S"+i . From a partial solution to this problem we deduce a generalization of the theorem of Smorynski and Stavi on cofinal extensions of recursively saturated models of arithmetic.

ABSTRACT. We observe that the classification problem for countable models of arithmetic is Borel complete. On the other hand, the classification problems for finitely generated models of arithmetic and for recursively saturated models of arithmetic are Borel; we investigate the precise complexity of each of these. Finally, we show that the classification problem for pairs of recursively saturated models and for automorphisms of a fixed recursively saturated model are Borel complete.