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Utilizing sheaves over topological spaces as natural models, we expose the principles of a logic of extended and variable structures. The resulting model theory, particular case of the logic of topoi, reveals interesting connections between logic and geometry. In this context, we present a notion of generic structure over a sheaf which illuminates the relations between intuitionistic and classical logic, and unifies the fundamental results of first order logic, infinitary logic, and set theory on model construction.
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