Fredrik Engström
Research interests
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InterestsModels of Peano Arithmetic, Dependence Logic, Logical Constants, Logical reasoning
Publications
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Characterizing Quantifier Extensions of Dependence Logic
02/2012;
We characterize the expressive power of extensions of Dependence Logic and Independence Logic by monotone generalized quantifiers in terms of quantifier extensions of existential second-order logic.... [more] We characterize the expressive power of extensions of Dependence Logic and Independence Logic by monotone generalized quantifiers in terms of quantifier extensions of existential second-order logic.
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Generalized quantifiers in Dependence Logic
03/2011;
We introduce generalized quantifiers, as defined in Tarskian semantics by Mostowski and Lindstr\"om, in logics whose semantics is based on teams instead of assignments, e.g., IF-logic and Dependence logic. Both the monotone and the non-monotone case is considered. It is argued that to handle qu... [more] We introduce generalized quantifiers, as defined in Tarskian semantics by Mostowski and Lindstr\"om, in logics whose semantics is based on teams instead of assignments, e.g., IF-logic and Dependence logic. Both the monotone and the non-monotone case is considered. It is argued that to handle quantifier scope dependencies of generalized quantifiers in a satisfying way the dependence atom in Dependence logic is not well suited and that the multivalued dependence atom is a better choice. This atom is in fact definably equivalent to the \emph{independence atom} recently introduced by V\"a\"an\"anen and Gr\"adel.
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Non-permutation invariant Borel quantifiers
03/2010;
Every permutation invariant Borel subset of the space of countable structures is definable in $\La_{\omega_1\omega}$ by a theorem of Lopez-Escobar. We prove variants of this theorem relative to fixed relations and fixed non-permutation invariant quantifiers. Moreover we show that for every closed su... [more] Every permutation invariant Borel subset of the space of countable structures is definable in $\La_{\omega_1\omega}$ by a theorem of Lopez-Escobar. We prove variants of this theorem relative to fixed relations and fixed non-permutation invariant quantifiers. Moreover we show that for every closed subgroup $G$ of the symmetric group $S_{\infty}$, there is a closed binary quantifier $Q$ such that the $G$-invariant subsets of the space of countable structures are exactly the $\La_{\omega_1\omega}(Q)$-definable sets. Comment: 10 pages
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A note on standard systems and ultrafilters
04/2008;
Let $(M,\scott X) \models \ACA$ be such that $P_\scott X$, the collection of all unbounded sets in $\scott X$, admits a definable complete ultrafilter and let $T$ be a theory extending first order arithmetic coded in $\scott X$ such that $M$ thinks $T$ is consistent. We prove that there is an end-ex... [more] Let $(M,\scott X) \models \ACA$ be such that $P_\scott X$, the collection of all unbounded sets in $\scott X$, admits a definable complete ultrafilter and let $T$ be a theory extending first order arithmetic coded in $\scott X$ such that $M$ thinks $T$ is consistent. We prove that there is an end-extension $N \models T$ of $M$ such that the subsets of $M$ coded in $N$ are precisely those in $\scott X$. As a special case we get that any Scott set with a definable ultrafilter coding a consistent theory $T$ extending first order arithmetic is the standard system of a recursively saturated model of $T$. Comment: 8 pages. To appear in the Journal of Symbolic Logic
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Expansions, omitting types, and standard systems
11/2004;
Recursive saturation and resplendence are two important notions in models of arithmetic. Kaye, Kossak, and Kotlarski introduced the notion of arithmetic saturation and argued that recursive saturation might not be as rigid as first assumed. In this thesis we give further examples of variations of re... [more] Recursive saturation and resplendence are two important notions in models of arithmetic. Kaye, Kossak, and Kotlarski introduced the notion of arithmetic saturation and argued that recursive saturation might not be as rigid as first assumed. In this thesis we give further examples of variations of recursive saturation, all of which are connected with expandability properties similar to resplendence. However, the expandability properties are stronger than resplendence and implies, in one way or another, that the expansion not only satisfies a theory, but also omits a type. We conjecture that a special version of this expandability is in fact equivalent to arithmetic saturation. We prove that another of these properties is equivalent to \beta-saturation. We also introduce a variant on recursive saturation which makes sense in the context of a standard predicate, and which is equivalent to a certain amount of ordinary saturation. The theory of all models which omit a certain type p(x) is also investigated. We define a proof system, which proves a sentence if and only if it is true in all models omitting the type p(x). The complexity of such proof systems are discussed and some explicit examples of theories and types with high complexity, in a special sense, are given. We end the thesis by a small comment on Scott's problem. We prove that, under the assumption of Martin's axiom, every Scott set of cardinality <2^{\aleph_0} closed under arithmetic comprehension which has the countable chain condition is the standard system of some model of PA. However, we do not know if there exists any such uncountable Scott sets.
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Satisfaction classes in nonstandard models of first-order arithmetic
10/2002;
A satisfaction class is a set of nonstandard sentences respecting Tarski's truth definition. We are mainly interested in full satisfaction classes, i.e., satisfaction classes which decides all nonstandard sentences. Kotlarski, Krajewski and Lachlan proved in 1981 that a countable model of PA adm... [more] A satisfaction class is a set of nonstandard sentences respecting Tarski's truth definition. We are mainly interested in full satisfaction classes, i.e., satisfaction classes which decides all nonstandard sentences. Kotlarski, Krajewski and Lachlan proved in 1981 that a countable model of PA admits a satisfaction class if and only if it is recursively saturated. A proof of this fact is presented in detail in such a way that it is adaptable to a language with function symbols. The idea that a satisfaction class can only see finitely deep in a formula is extended to terms. The definition gives rise to new notions of valuations of nonstandard terms; these are investigated. The notion of a free satisfaction class is introduced, it is a satisfaction class free of existential assumptions on nonstandard terms. It is well known that pathologies arise in some satisfaction classes. Ideas of how to remove those are presented in the last chapter. This is done mainly by adding inference rules to M-logic. The consistency of many of these extensions is left as an open question.
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Expansions, omitting types, and standard systems /
Abstract sheet laid in. Thesis (Ph. D.)--Chalmers University of Technology, 2004.
Following (5)
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David Mccallum
Göteborgs universitet -
Johan Wessberg
Göteborgs universitet -
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Giuseppe Primiero
University of Ghent -
Sam Sanders
University of Ghent
