Skolem's Paradox
978-613-1-16804-8
6131168040
88
2010-08-10
34.00 €
eng
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematical logic and philosophy, Skolem's paradox is a seeming contradiction that arises from the downward Löwenheim-Skolem theorem. Thoralf Skolem (1922) was the first to discuss the seemingly contradictory aspects of the theorem, and to discover the relativity of set-theoretic notions now known as non-absoluteness. Although it is not an actual antinomy like Russell's paradox, the result is typically called a paradox, and was described as a "paradoxical state of affairs" by Skolem (1922: p. 295). Skolem's paradox is that every countable axiomatisation of set theory in first-order logic, if it is consistent, has a model that is countable. This appears contradictory because it is possible to prove, from those same axioms, a sentence which intuitively says that there exist sets that are not countable. Thus the seeming contradiction is that a model which is itself countable, and which contains only countable sets, satisfies the first order sentence that intuitively states "there are uncountable sets".
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