Positive model theory and infinitary logic

Abstract: We study the basic properties of a dual "spectral" topology on positive type spaces of h-inductive theories and its essential connection to infinitary logic. The topology is Hausdorff, has the Baire property, and its compactness characterises positive model completeness; it also has a basis of clopen sets and is described by the formulas of geometric logic. The "geometric types" are closed in the type spaces under all the operations of infinitary logic, and we introduce a positive analogue of existentially universal structures, through which we interpret the full first order logic in positive type spaces. This shows how "positive $\omega$-saturation" is a fundamental connection between positive and infinitary logic, and we suggest a geometric analogue of positive Morleyisation.