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**11 - 16**of**16**### ANALYTIC STRUCTURES AND MODEL THEORETIC COMPACTNESS

"... In recent years, there has been considerable interest in replicating the successful development of first-order model theory in non first-order contexts. One such context that has emerged as an important test case is that of metric structures, i.e., structures whose sorts are metric spaces. The appea ..."

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In recent years, there has been considerable interest in replicating the successful development of first-order model theory in non first-order contexts. One such context that has emerged as an important test case is that of metric structures, i.e., structures whose sorts are metric spaces. The appeal of a rich model theory

### Algorithmic issues of the Feferman–Vaught Theorem

, 2004

"... The classical Feferman–Vaught Theorem for First Order Logic explains how to compute the truth value of a first order sentence in a generalized product of first order structures by reducing this computation to the computation of truth values of other first order sentences in the factors and evaluatio ..."

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The classical Feferman–Vaught Theorem for First Order Logic explains how to compute the truth value of a first order sentence in a generalized product of first order structures by reducing this computation to the computation of truth values of other first order sentences in the factors and evaluation of a monadic second order sentence in the index structure. This technique was later extended by Läuchli, Shelah and Gurevich to monadic second order logic. The technique has wide applications in decidability and definability theory. Here we give a unified presentation, including some new results, of how to use the Feferman– Vaught Theorem, and some new variations thereof, algorithmically in the case of Monadic Second Order Logic MSOL. We then extend the technique to graph polynomials where the range of the summation of the monomials is definable in MSOL. Here the Feferman–Vaught Theorem for these polynomials generalizes well known splitting theorems for graph polynomials. Again, these can be used algorithmically. Finally, we discuss extensions of MSOL for which the Feferman–Vaught Theorem holds as well.

### and

, 2001

"... Let HC ′ denote the set of sets of hereditary cardinality less than 2ω. We consider reflection principles for HC ′ in analogy with the Levy reflection prin-ciple for HC. Let B be a class of complete Boolean algebras. The principle Max(B) says: If R(x1,..., xn) is a property which is provably persist ..."

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Let HC ′ denote the set of sets of hereditary cardinality less than 2ω. We consider reflection principles for HC ′ in analogy with the Levy reflection prin-ciple for HC. Let B be a class of complete Boolean algebras. The principle Max(B) says: If R(x1,..., xn) is a property which is provably persistent in ex-tensions by elements of B, then R(a1,..., an) holds whenever a1,..., an ∈ HC ′ and R(a1,..., an) has a positive IB-value for some IB ∈ B. Suppose C is the class of Cohen algebras. We prove that Con(ZF) implies Con(ZFC+Max(C)). For a different principle, let CCC be the class of all CCC algebras. We prove that ZF+ Levy schema, and ZFC+Max(CCC) are equiconsistent. Max(CCC) implies MA, while Max(C) implies ¬MA. We give applications of these reflection principles to Löwenheim-Skolem theorems of extensions of first order logic. For example, Max(C) implies that the Löwenheim number of the extension of first order logic by the Härtig quantifier is less than 2ω. 1