Simple theories
The authors study simple theories in the sense of \textit{S. Shelah} [Ann. Math. Logic 19, 177-203 (1980; Zbl 0489.03008)]. The Independence Theorem over a model \(M\) is the following: For \(A\supset M\), \(B\supset M\), \(A\) independent from \(B\) over \(M\), \(p\in S(M)\), and \(p_1\in S(A)\), \(p_2\in S(B)\) both nonforking extensions of \(p\) [in the sense of \textit{S. Shelah}, Classification theory and the number of non-isomorphic models. 2nd rev. ed. (1990; Zbl 0713.03013)], then there is \(q\in S(A\cup B)\) which extends both \(p_1\) and \(p_2\) and is also a nonforking extension of \(p\). The authors prove the Independence Theorem for the models of simple theories. Moreover, they prove the following: Theorem. Let \(T\) be an arbitrary theory. Then \(T\) is simple iff \(T\) has a notion of independence \(\Gamma\) which satisfies the Independence Theorem over a model. Moreover, for such a \(\Gamma\) holds: for all \(a\), \(B\), \(A\), \((a,B,A)\in\Gamma\) iff \(tp(a/B)\) does not fork over \(A\). The Independence Theorem is also proved for Lascar strong types. Various notions of rank and their interconnections are considered.
scientific article; zbMATH DE number 1109922
scientific article