Continuity and Logical Completeness. An application of sheaf theory and

Continuity and Logical Completeness. An application of sheaf theory and topoi. section: Category theory 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 in topos theorey are then discussed in this setting, and the new logical completeness theorems of [1, 2, 3] for systems of higher-order logic are elucidated. The main argument can be outlined 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. Higher-order logic (HOL) can be presented algebraically. As a consequence of this fact, it has continuously variable models. 3. Variable models are classical mathematical objects; namely, sheaves. 4. HOL is complete with respect to such continuously variable models. Standard semantics appears thereby as the constant case of " no variation. " In this sense, HOL is the logic of continuous variation. This argument is developed in four sections: (i) the algebraic formulation of HOL is given; (ii) rings of real-valued functions are considered as an example of variable structure; (iii) the idea of continuously variable sets is then discussed; and finally, (iv) it is explained how HOL is the logic of continuous variation.