CONSISTENCY is an essential property of any mathematical system. Without it two conflicting statements may be deduced within the system. Independence, on the other hand, is not considered essential, and, indeed, it is sometimes not desirable. If the set of postulates in a high school geometry text were independent, then the first few basic theorems would be so difficult that virtually no high school student could understand their proofs. At the foundations level, however, it is considered aesthetically desirable that the postulates of a system be independent. It is generally not an easy matter to prove that a set of postulates is independent.1 It is believed that when Euclid stated his five postulates for geometry he thought that his parallel postulate was demonstrable from the other postulates, but he was unable to give a valid proof of it. Mathematicians sought a proof for two thousand years, but it has been only within the last century or so that the search has been shown to be futile. It took two thousand years and the development of non-Euclidean geometries to show that the parallel postulate is independent.
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