NECESSARY AND SUFFICIENT CONDITION FOR THE EXISTENCE OF INTERNAL TIME ON AN ORIENTED SET

Abstract

The notion of oriented set is the most elementary technical notion of the theory of changeable sets, which is needed for the general definition of changeable set notion. The main motivation for building the theory of changeable sets was the sixth Hilbert problem, that is, the problem of mathematically rigorous formulation of the fundamentals of theoretical physics. From the formal point of view oriented set is the simplest relation system with one reflexive binary relation. Oriented sets may be interpreted as simplest abstract models of sets of changing objects, evolving in the framework of the single (specified) reference frame. From the other hand in the framework of oriented sets we can give the mathematically strict and abstract definition of the notion of time as some mapping from some linearly ordered set to the power set of the set of elementary states of oriented set. Internal time may be considered as most natural time for an oriented set. From intuitive point of view internal time is the time, which can be “observed from the inside” of the oriented set. In the present paper we solve the problem of the existence of internal time on an oriented set without any synchronization. We prove necessary and sufficient condition for the existence of such time.