Logic1130 - Last time, We proved the completeness theorem...

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Unformatted text preview: Last time, We proved the completeness theorem for 1st-order logic by constructing a sys- tematic tableau. If a formula X is valid (or FX is not satis- fiable), then it is provable by a systematic tableau. Ex. HW #9, Q2. Lwenheims theorem follows as a corol- lary: If X is satisfiable, then X is satisfiable in a Instead of a formula, consider a de- numerable set of formulas: S = {X1, X2, , Xn, }. Wed like to prove the same theorem for S: If S is satisfiable, then S is satis- fiable in a denumerable domain (c- alled Skolem-Lwenheims theorem). A systematic tableau can be constructed in the following way. Start the tableau with X1 at the origin (1st stage). If all finite subsets of S are satisfiable, then for every n, the set {X1, X2, , Xn} is satisfiable (Problem 17.3). Therefore, the tableau will not close at any stage, and there will be an infinite branch (by Knigs lemma) which must be open. is a Hintikka set (every formula is fulfilled p.210) and it contains all elements of S (since weve tacked every Xn to it). is satisfiable in a denumerable domain (by Hintikkas lemma). Hence we just proved:...
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This note was uploaded on 02/04/2010 for the course HSS Hss105 taught by Professor Yeelee during the Fall '09 term at Korea Advanced Institute of Science and Technology.

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Logic1130 - Last time, We proved the completeness theorem...

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