Topoi and Categories of Fuzzy Sets

TL;DR: Any topos can be considered as a generalization of model theory, mostly for theorems which originally have been intended for interpretation in classical set valued models only.

Abstract: Motivation. In Chapters 2 and 3 we have introduced the notion of a topos and showed how classical many sorted logic can be interpreted in a topos. This fact has a very natural interpretation: by interpreting the logic we can define, in fact, a model of some theory. The most frequently used theory is that of classical (sometimes only intuitive) set theory. It is well known that for construction of (also classical) models of set theory we use again model theory (of some different type) and any other model of set theory we build up in scopes of this another set theory. From this point of view, any topos can be considered as a generalization of model theory, mostly for theorems which originally have been intended for interpretation in classical set valued models only. The principal advantages of this generalized model theory are the following: First, the internal logic of this interpretation is not Boolean (in general) and it follows that the differences between results obtained by interpreting theorems in topoi and sets, respectively, are more substantial than formal. Second, by interpreting formulas and theorems in topoi we can obtain objects in these categories with some very special additional properties, which cannot be possible to construct simply in classical set valued models. For example, both these advantages have been used to find some counterexamples of the well known hypotheses in the set theory.