Lindenbaum's lemma
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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]
Notes
- ↑ Tarski, A. On Fundamental Concepts of Metamathematics, 1930.
References
- Crossley, J.N.; Ash, C.J.; Brickhill, C.J.; Stillwell, J.C.; Williams, N.H. (1972). What is mathematical logic?. London-Oxford-New York: Oxford University Press. p. 16. ISBN 0-19-888087-1.
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