Topological Model Theory Research Articles

Some Connections between Topological and Modal Logic

AbstractWe study modal logics based on neighbourhood semantics using methods and theorems having their origin in topological model theory. We thus obtain general results concerning completeness of modal logics based on neighbourhood semantics as well as the relationship between neighbourhood and Kripke semantics. We also give a new proof for a known interpolation result of modal logic using an interpolation theorem of topological model theory.

Stability theory for topological logic, with applications to topological modules

In this paper I show how to develop stability theory within the context of the topological logic first introduced by McKee [Mc 76], Garavaglia [G 78] and Ziegler [Z 76]. I then study some specific applications to topological modules; in particular I prove two quantifier élimination theorems, one a generalization of a result of Garavaglia.In the first section I present a summary of basic results on topological model theory, mostly taken from the book of Flum and Ziegler [FZ 80]. This is done primarily to fix notation, but I also introduce the notion of anLt-elementary substructure. The important point with this concept, as with many others, appears to be to allow only individuals to appear as parameters, not open sets.In the second section I begin the study of stability theory forLt. I first develop a translation of the topological languageLtinto an ordinary first-order languageL*. The first main theorem is (2.3), which shows that the translation is faithful to the model-theoretic content ofLt, and provides the necessary tools for studyingLttheories in the context of ordinary first-order logic. The translation allows me to considerindividual stability theoryforLt: the stability-theoretic study of those types ofLtin which only individual variables occur freely and in which only individuals occur as parameters. I originally developed this stability theory entirely withinLt; the fact that the theorems and their proofs were virtually identical to those in ordinary first order logic suggested the reduction fromLttoL*.

Topological model theory with an interior operator: Consistency properties and back — and forth arguments

The logicL(I) is introduced. Compactness and Lowenheim-Skolem theorems for topological models are proved. Several axiomatizations are given and proved to be complete. Interpolation- and preservation theorems are proved using consistency properties. A Back and Forth criterion for elementary equivalence is proved. Most of the results are extended (appropriately) to infinitary logic.

A note on topological model theory

A note on topological model theory