Quantifiers and Sheaves

The unity of opposites in the title is essentially that between logic and geometry, and there are compelling reasons for maintaining that geometry is the leading aspect. At the same lime, in the present joint work with Myles Tierney there are important influences in the other direction: a Grothendieck " topology " appears most naturally as a modal operator, of the nature " it is locally the case that ", the usual logical operators such as V, 3, => have natural analogues which apply to families of geometrical objects rather than to propositional functions, and an important technique is to lift constructions first understood for " the " category S of abstract sets to an arbitrary topos. We first sum up the principal contradictions of the Grothendieck-GiraudVerdier theory of topos in terms of four or five adjoint functors, significantly generalizing the theory to free it of reliance on an external notion of infinite limit (in particular enabling one to claim that in a sense logic is a special case of geometry). The method thus developing is then applied to intrinsically define the concept of Boolean-valued model for S (BVM/S) and to prove the independence of the continuum hypothesis free of any use of transfinite induction. The second application of the method outlined here is an intrinsic geometric construction of the Chevalley-Hakim global spectrum of a ringed topos free of any choice of a " site of definition ".