# completeness

## EnglishEdit

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### NounEdit

completeness (usually uncountable, plural completenesses)

1. the state or condition of being complete
2. (logic) The property of a logical theory that whenever a wff is valid then it must also be a theorem. Symbolically, letting T represent a theory within logic L, this can be represented as the property that whenever ${\displaystyle T\vDash \phi }$  is true, then ${\displaystyle T\vdash \phi }$  must also be true, for any wff φ of logic L.
• 2002, Stephen Cole Kleene, Mathematical Logic, Mineola, New York: Dover Publications, ISBN 0-486-42533-9 (pbk.) Invalid ISBN, page 314:
THEOREM 37°. (Gödel's completeness theorem 1930.) In the predicate calculus H:
(a) If ${\displaystyle \vDash F}$  [or even if ${\displaystyle \aleph _{0}}$ -${\displaystyle \vDash F}$ ], then ${\displaystyle \vdash F}$ . If ${\displaystyle E_{1},...,E_{k}\vDash F}$  [or even if ${\displaystyle E_{1},...,E_{k}\ \aleph _{0}}$ -${\displaystyle \vDash F}$ ], then ${\displaystyle E_{1},...,E_{k}\vdash F}$ .
(b) [...]