Consistency and Independence in Postulational Technique

Extract

Despite the skepticism of many mathematicians and logicians as to the possibility of any test which will show conclusively the consistency or independence of the members of a postulate set, several methods have nevertheless been devised and employed, e.g., the empirical methods of Russell and Huntington, the internal method of Hilbert, and the reflective method of Royce. However, with the possible exception of Hilbert's method (which is still available only in the form of fragments and suggestions), these techniques require us to forsake the purely formal or abstract mode of analysis, and instead to rely in some sense on “concrete representations” or “interpretations” or “modes of action” which are said to “satisfy” the postulates in question. In short, they are all interpretational methods—and here is the crux of the problem. “Is it possible,” queries Weiss, “that the only way we can determine whether a set is consistent is by seeing all the postulates actually exemplified in some object? If so, we must arbitrarily assume that the object is self-consistent, so that the proof of consistency must ultimately rest on a dogma. As independence rests on consistency there are therefore no satisfactory proofs as yet of either independence or consistency.” The question “is thus answered,” according to Young, “only by reference to a concrete representation of the abstract ideas involved, and it is such concrete representations that we wished especially to avoid. At the present time, however, no absolute test for consistency is known.” Moreover, “suppose,” says the late Dr. Eaton, “that the system has no interpretation: how can the consistency (and) independence … of the postulates be shown? There should be some analytic way—purely in the realm of the abstract, without interpretation—of establishing these properties of a set of postulates. This is an important problem that awaits solution.”