Lindenbaum's lemma

In mathematical logic, Lindenbaum's lemma, named after Adolf Lindenbaum, 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.

Uses

It is used in the proof of Gödel's completeness theorem, among other places.^{[citation needed]}

Extensions

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.

History

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