Logical paradoxes and their role in mathematical modeling

A short explanation of essence and history of logical paradoxes and of contradictory models of optimization and classification is given. Paradoxes are considered in set theory, the cause of which is to use the concept of actual infinity and transferring methods suitable for finite sets on the endless sets. Some logical paradoxes are associated with improper predicate systems, i.e. such inconsistent systems of predicates which only improper object can be put in compliance. The way of analysis of these paradoxes is considered which consists of the expansion of existing ideas about objects in easing of the requirements imposed while determining of the object, in the expansion of the meaning of the concept of “existence”. We consider the modeling of objects with incompatible systems of linear inequalities. The solution of contradictory systems is proposed on the way of fuzzy conceptions and the collective solutions (this can be considered as modeling of consultation). In the latter case more individual approach to resolve the paradoxes is investigated. Some means of easing of the requirements of certain absolute criteria for solving the problem are used. The case of the analysis of non-formalized and even non-formalizable problems is especially important. Proposed approaches to non-formalizable models were discussed with N.N. Nepeivoda within the World Congress of the science logic.