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# Logic1130 - axiom systems IF 1 2 3 Skolem-Lwenheims theorem...

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] F° °°° °° axiom systems 1. Skolem-Löwenheim’s theorem 2. What is an axiom system? – Euclid’s case 3. Axiom systems in logic

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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. Löwenheim’s theorem follows as a corol- lary: If X is satisfiable, then X is satisfiable in a denumerable domain.
Instead of a formula, consider a de- numerable set of formulas: S = {X1, X2, …, Xn, …}. We’d like to prove the same theorem for S: If S is satisfiable, then S is satis- fiable in a denumerable domain (c- alled Skolem-Löwenheim’s theorem). A systematic tableau can be constructed in the following way. Start the tableau with X1 at the origin (1st stage).

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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 Θ (by König’s lemma) which must be open. Θ
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Logic1130 - axiom systems IF 1 2 3 Skolem-Lwenheims theorem...

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