ABSTRACT
This chapter considers some set-theoretical questions concerning the existence and uniqueness of solutions of ordinary differential equations. It deals with those ordinary differential equations of first order, the right-hand sides of which are nonmeasurable in the Lebesgue sense. The existence of a solution of a first-order ordinary differential equation with a continuous right-hand side is stated by the famous Peano theorem. The chapter shows that this classical theorem does not rely on any form of the Axiom of Choice and, in fact, is a result of ZF set theory. It also proves that the existence and uniqueness of a solution can be fulfilled even for some ordinary differential equations whose right-hand sides are nonmeasurable in the Lebesgue sense.
