In which the fundamentals of RUSS are presented within a framework constructed from BISH and an axiom derived from Church's thesis. The notion of a programming system is introduced in Section 1, and is used in discussions of omniscience principles, continuity, and the intermediate value theorem. Section 3 deals with Specter's remarkable construction of a bounded increasing sequence of rational numbers that is eventually bounded away from any given (recursive) real number. The next section contains a strong counter-example to the Heine-Borel theorem, and a discussion of the Lebesgue measure of the unit interval and its compact subsets. The last two sections of the chapter deal with Ceitin's theorem on the continuity of real-valued functions.