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Lindenbaum's lemma

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Title: Lindenbaum's lemma  
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Subject: Mathematical Logic, Adolf Lindenbaum, Theory (mathematical logic), Lemmas, Logic Gates & Human Logic
Collection: Lemmas, Mathematical Logic
Publisher: World Heritage Encyclopedia

Lindenbaum's lemma

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.


  • Uses 1
  • Extensions 2
  • History 3
  • Notes 4
  • References 5
  • External links 6


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.


  • Crossley, J.N.; Ash, C.J.; Brickhill, C.J.; Stillwell, J.C.; Williams, N.H. (1972). What is mathematical logic?. London-Oxford-New York:  

External links

  • University of Texas, A Causal Theory of Modal Knowledge (Including Logical and Mathematical Knowledge
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