Oriented sets are the simplest mathematical structures that model collections of evolving objects. This work is devoted to the problem of the existence of internal time on a synchronized oriented set. From an intuitive point of view, internal time is the time whose flow can be \textquotedbl observed and recorded\textquotedbl{} while \textquotedbl living inside\textquotedbl{} the oriented set. In the paper we prove that the sufficient condition for the existence and uniqueness of internal time on a synchronized oriented set, obtained in previous works, is not a necessary one. Also we establish some
necessary condition for the existence of internal time on a synchronized oriented set and show that this condition is not sufficient.

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