The Craig Interpolation Theorem in abstract model theory

Väänánen, Jouko

doi:10.1007/s11229-008-9357-z

Cited by 4 publications

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“…Since our logic is not compact, it could be the case that some of these properties hold for ε-logic, while others do not. For a discussion of these properties for various extensions of first-order logic see for example Väänänen [23].…”

Section: Future Researchmentioning

confidence: 99%

Model Theory of Measure Spaces and Probability Logic

Kuyper

Terwijn

2013

The Review of Symbolic Logic

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We study the model-theoretic aspects of a probability logic suited for talking about measure spaces. This nonclassical logic has a model theory rather different from that of classical predicate logic. In general, not every satisfiable set of sentences has a countable model, but we show that one can always build a model on the unit interval. Also, the probability logic under consideration is not compact. However, using ultraproducts we can prove a compactness theorem for a certain class of weak models.

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Section: Future Researchmentioning

confidence: 99%

Model Theory of Measure Spaces and Probability Logic

Kuyper

Terwijn

2013

The Review of Symbolic Logic

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“…None of these theorems is true for the classical L κκ logic, by the work of Gostanian and Hrbaček ([GoHr]) and Malitz ([Mal]). A discussion involving many of these developments can be found in [Vä1]. Our paper can be seen as an extension of Scott's analysis of countable models to chain models of size κ.…”

Section: Introductionmentioning

confidence: 97%

Chain Models, Trees of Singular Cardinality and Dynamic Ef-Games

Džamonja

Väänánen

2011

J. Math. Log.

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Let κ be a singular cardinal. Karp's notion of a chain model of size κ is defined to be an ordinary model of size κ along with a decomposition of it into an increasing union of length cf (κ). With a notion of satisfaction and (chain)-isomorphism such models give an infinitary logic largely mimicking first order logic. In this paper we associate to this logic a notion of a dynamic EF-game which gauges when two chain models are chain-isomorphic. To this game is associated a tree which is a tree of size κ with no κ-branches (even no cf (κ)-branches). The measure of how non-isomorphic the models are is reflected by a certain order on these trees, called reduction. We study the collection of trees of size κ with no κ-branches under this notion and prove that when cf (κ) = ω this collection is rather regular; in particular it has universality number exactly κ+. Such trees are then used to develop a descriptive set theory of the space cf (κ)κ. The main result of the paper gives in the case of κ strong limit singular an exact connection between the descriptive set-theoretic complexity of the chain isomorphism orbit of a model, the reduction order on the trees and winning strategies in the corresponding dynamic EF games. In particular we obtain a neat analog of the notion of Scott watershed from the Scott analysis of countable models.

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Craig Interpolation for Decidable First-Order Fragments

ten Cate,

Comer

2024

Lecture Notes in Computer Science

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We show that the guarded-negation fragment (GNFO) is, in a precise sense, the smallest extension of the guarded fragment (GFO) with Craig interpolation. In contrast, we show that the smallest extension of the two-variable fragment ($$\textrm{FO}^2 $$ FO 2 ), and of the forward fragment (FF) with Craig interpolation, is full first-order logic. Similarly, we also show that all extensions of $$\textrm{FO}^2 $$ FO 2 and of the fluted fragment (FL) with Craig interpolation are undecidable.

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The Craig Interpolation Theorem in abstract model theory

Cited by 4 publications

References 50 publications

Model Theory of Measure Spaces and Probability Logic

Model Theory of Measure Spaces and Probability Logic

Chain Models, Trees of Singular Cardinality and Dynamic Ef-Games

Craig Interpolation for Decidable First-Order Fragments

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