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. The lemma is a special case of the ultrafilter lemma for Boolean algebras, applied to the Lindenbaum algebra of a theory.


It is used in the proof of Gödel's completeness theorem, among other places.


The effective version of the lemma's statement, "every consistent computably enumerable theory can be extended to a complete consistent computably enumerable theory," fails (provided Peano Arithmetic is consistent) by Gödel's incompleteness theorem.


The lemma was not published by Adolf Lindenbaum; it is originally attributed to him by Alfred Tarski.[1]


  1. ^ Tarski, A. On Fundamental Concepts of Metamathematics, 1930.