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We say a mathematical theory is decidable if there is an effective method of determining the validity of each statement of the theory. If there is no such method, the theory is undecidable. It is clear that if there is a mechanical way of transforming each statement of an undecidable theory into an equivalent statement of another theory, the second theory is also undecidable. This principle, together with the fact that the arithmetic of natural numbers is undecidable, enables us to solve the decision problem for fields of finite degree over the rationals. (en) |
