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A SYSTEM (K, R), where K is a class of elements Ay B, C9 • • • and Ris a dyadic relation, is called a roell ordered system when the following conditions are satisfied:* (a) the system (K, R) is a series; and (b) every subsystem of (K, R) has a leading element.] Now when condition (b) is added to the conditions (a) which define a series, some of the conditions (a) become redundant. After eliminating these redundancies, we find the following three sets of independent postulates for well ordered systems, each of these three sets being in fact " completely independent " in the sense of E. H. Moore. (The numbering of the postulates is made to conform with that in the preceding note.)

Complete existential theory of the postulates for well ordered sets