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Continuity and Logical Completeness: An application of sheaf theory and topoi
, 2000
"... 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, realvalued functions on a topologic ..."
Abstract

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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, realvalued 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 nonspecialist. Some recent results are mentioned, including a new completeness theorem for higherorder logic. The main argument of this paper is as follows: 1. The distinction between the Particular and the Abstract General is present in that between the Constant and the Continuously Variable. More specially, continuous variation is a form of abstraction. 2. Higherorder logic (HOL) can be presented algebraically. As a co...