Continuity and logical completeness: an application of sheaf theory and topoi.
The notion of a continuously variable quantity can be regarded as a generalization of that of a particular (constant) quantity, and the properties of such quantities are then akin to, and derived from, the properties of constants. For example, the continuous, real-valued functions on a topological space behave like the field of real numbers in many ways, but instead form a ring. Topos theory permits one to apply this same idea to logic, and to consider continuously variable sets (sheaves). In this expository paper, such applications are explained to the non-specialist. Some recent results are mentioned, including a new completeness theorem for higher-order logic.
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