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Diaconescu's Theorem
In mathematical logic, Diaconescu's theorem, or the Goodman–Myhill theorem, states that the full axiom of choice is sufficient to derive the law of the excluded middle, or restricted forms of it, in constructive set theory. It was discovered in 1975 by Radu Diaconescu and later by Goodman and Myhill. Already in 1967, Errett Bishop posed the theorem as an exercise (Problem 2 on page 58 in ''Foundations of constructive analysis''E. Bishop, ''Foundations of constructive analysis'', McGraw-Hill (1967)). Proof For any proposition P\,, we can build the sets : U = \ and : V = \. These are sets, using the axiom of specification. In classical set theory this would be equivalent to : U = \begin \, & \mbox P \\ \, & \mbox \neg P\end and similarly for V\,. However, without the law of the excluded middle, these equivalences cannot be proven; in fact the two sets are not even provably finite (in the usual sense of being in bijection with a natural number, though they would be in the De ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |