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 eld 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. 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. 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. 1 The argument will be 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 nally, (iv) it is explained how HOL is the logic of continuous variation 1 Algebraic logic Categorical logic can be seen as the successful completion of the program of \algebraicizing" logic begun in the 19-century. Everyone is familiar with the boolean algebra approach to propositional logic, but the treatment of quan-tiication in particular has posed a serious obstacle to extending the algebraic treatment. The categorical treatment of quantiiers as adjoint functors|due to F.W. Lawvere in the 1960s|solved this problem, although it has been little appreciated until very recently. Category theory is of course a branch of abstract algebra, but the sense in which the categorical treatment of logic is \algebraic" is deeper than just that. Rather, it is the recognition of the quantiiers|and indeed all of the logical operations|as adjoint functors that makes logic algebraic. For it is a general fact about adjoints that they always admit an algebraic description, in a deenite, technical sense. …