U-sets as a possibilistic set theory

The motivation of possibility theory is to represent the imprecision intrinsic in natural language, in particular in expressions of the form xisR. A suitable topos of presheaves, i.e. an extension of classical set theory, is presented where sets of the form {x: xisR} can be constructed with x is R having the meaning given in possibility theory. The aim of the paper is to give an explicit mathematical description of the context sensitivity of linguistic imprecision. This can be done by defining sets that vary over the possible contexts, i.e. sets in a topos of presheaves defined over a category that represents the flow of information