Lindenbaum's lemma
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In mathematical logic, Lindenbaum's lemma states that any consistent theory of predicate logic can be extended to a complete consistent theory. It is used in the proof of Gödel's completeness theorem, among other places. The lemma is a special case of the ultrafilter lemma for Boolean algebras, applied to the Lindenbaum algebra of a theory.
The effective version of the lemma's statement, "every consistent computably enumerable theory can be extended to a complete consistent computably enumerable theory," fails by Gödel's incompleteness theorem.
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