Incompleteness theorem

Japanese: 不完全性定理 - ふかんぜんせいていり

A system is said to be complete when it is possible to determine the truth or falsity of all propositions that can be formulated within that system. The incompleteness theorems state that if a formal system including natural number theory is consistent, then it is not complete (Gödel's First Incompleteness Theorem), and that if a formal system including natural number theory is consistent, then that consistency cannot be derived from within that system (Gödel's Second Incompleteness Theorem). It was announced by Gödel in 1931. Gödel proved the First Incompleteness Theorem under the assumption of omega-consistency, which is stronger than consistency, but later J.B. Rosser proved that the First Incompleteness Theorem holds even if omega-consistency is replaced with consistency, and so it has taken its current form.

Source : Heibonsha Encyclopedia About MyPedia Information

Japanese:

ある体系内で定式化できるあらゆる命題に対して，その真偽がすべて決定できるとき，その体系は完全といわれる。不完全性定理とは〈自然数論を含む形式的体系が無矛盾ならば，それは完全ではない〉（ゲーデルの第１不完全性定理），〈自然数論を含む形式的体系が無矛盾ならば，その無矛盾性をその体系内から導くことはできない〉（ゲーデルの第２不完全性定理）をいう。1931年ゲーデルが発表。ゲーデルは第１不完全性定理を無矛盾性よりも強いオメガ無矛盾という仮定のもとで証明したが，その後ロッサーJ.B.Rosserがオメガ無矛盾を無矛盾に置き換えても第１不完全性定理が成り立つことを証明し，現在の形となっている。

出典　株式会社平凡社百科事典マイペディアについて　情報

<<: Kingdom of Buganda - Buganda Kingdom (English spelling) Buganda

>>: Underemployment - underemployment