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Applications of the Compactness Theorem

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Ultraproducts

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Automated Theorem Proving and Axiomatic Mathematics

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Ordinals

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5. Soundness and Completeness - Logic for Beginners

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Roger Penrose explains Godel's incompleteness theorem in 3 minutes

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41. Logic. Gödel's Completeness theorem

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The paradox at the heart of mathematics: Gödel's Incompleteness Theorem - Marcus du Sautoy