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2.6.1: Introduction
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The completeness theorem is one of the most fundamental results about logic.
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2.6.2: Outline of the Proof
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The proof of the completeness theorem is a bit complex, and upon first reading it, it is easy to get lost. So let us outline the proof.
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2.6.3: Complete Consistent Sets of Sentences
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Complete sets of sentences leave no questions unanswered. For any sentence \(A\), \(\Gamma\) “says” if \(A\) is true or false.
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2.6.4: Henkin Expansion
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Part of the challenge in proving the completeness theorem is that the model we construct from a complete consistent set \(\Gamma\) must make all the quantified formulas in \(\Gamma\) true. In order to guarantee this, we use a trick due to Leon Henkin.
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2.6.5: Lindenbaum’s Lemma
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We now prove a lemma that shows that any consistent set of sentences is contained in some set of sentences which is not just consistent, but also complete.
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2.6.6: Construction of a Model
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Right now we are not concerned about \(=\), i.e., we only want to show that a consistent set \(\Gamma\) of sentences not containing \(=\) is satisfiable. We first extend \(\Gamma\) to a consistent, complete, and saturated set \(\Gamma^*\). In this case, the definition of a model \(M(\Gamma^*)\) is simple.
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2.6.7: Identity
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The construction of the term model given in the preceding section is enough to establish completeness for first-order logic for sets \(\Gamma\) that do not contain \(=\). It does not work, however, if \(=\) is present. We can fix this using a construction known as “factoring.”
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2.6.8: The Completeness Theorem
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Let’s combine our results: we arrive at the completeness theorem. Let \(\Gamma\) be a set of sentences. If \(\Gamma\) is consistent, it is satisfiable.
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2.6.9: The Compactness Theorem
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One important consequence of the completeness theorem is the compactness theorem. The compactness theorem states that if each finite subset of a set of sentences is satisfiable, the entire set is satisfiable—even if the set itself is infinite.
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2.6.10: A Direct Proof of the Compactness Theorem
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We can prove the Compactness Theorem directly, without appealing to the Completeness Theorem, using the same ideas as in the proof of the completeness theorem.
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2.6.11: The Löwenheim-Skolem Theorem
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The Löwenheim-Skolem Theorem says that if a theory has an infinite model, then it also has a model that is at most countably infinite.
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2.6.12: Summary
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