Lindenbaum's lemma

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In mathematical logic, Lindenbaum's lemma states that any consistent theory can be extended to a maximal consistent theory. A theory is a set of sentences closed under deduction (or semantic consequence). It is referred to as a 'lemma' because it often used as such in completeness proofs, e.g., of the first-order predicate calculus.

Adolf Lindenbaum proved numerous theorems, many of which may be aptly suited to the title 'Lindenbaum's lemma', but generally the term is reserved for the aforementioned theorem regarding consistent theories and their maximal extensions.