On Generic Stalks of Sheaves

Sheaf theoretic notions and methods have entered model theory during recent years and important constructions like reduced powers and products, unions of chains, Boolean valued models and models for non-classical logics appear here in a unified setting. If one looks at a sheaf as a generalized structure, then forcing becomes the genuine extension of the validity relation between models and formulae.! Stalks within a sheaf for which satisfaction and (weak) forcing coincide are customarily called (weakly) generic. Not every sheaf P has generic or weakly generic stalks but under some topological assumptions, e.g. local compactness, one has at least the existence of stalks which force a complete type of formulae. We call such stalks quasi-generic. For sheaves associated with pseudo-Boolean valued models, quasigeneric stalks are generic and a further specialization to diagram-sheaves then yields surprisingly simple proofs of Keisler's [4] important consistency results. The diagramsheaves which are defined in a first section will replace in our paper the classical " construction of models within the language by means of constants ". For any sheaf within an axiomatic class the quasi-generic stalks force complete sets of sentences. Closure under direct limits turns out to be a sufficient condition for an axiomatic class to allow such completions by sheaves. For better readability, some basic notions concerning sheaves and forcing are mentioned, but the reader is referred to Ellerman's [2] article for a systematic account. The author also would like to express his sincere thanks to the referee for his many valuable suggestions and useful comments.