Gödelova věta o úplnosti predikátové logiky

Gödelova věta o úplnosti predikátové logiky (také jen Gödelova věta o úplnosti či věta o úplnosti) je základní větou matematické logiky. Dává do souvislosti syntaktický pojem dokazatelnosti a sémantický pojem pravdivosti v modelu.

Znění

Větou o úplnosti se obvykle nazývá následující ekvivalence. Implikace zleva doprava se někdy nazývá věta o korektnosti.

Formule je dokazatelná v teorii T, právě když platí v každém modelu T.

Důsledky

Gödelova věta o úplnosti má zcela základní význam pro celou matematickou logiku. Vyplývá z ní mnoho důležitých tvrzení a vět, například:

Historie

Větu o úplnosti dokázal poprvé v roce 1929 Kurt Gödel, v současné době se však častěji uvádí důkaz podaný později Leonem Henkinem.

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