completeness

Etymology

complete + -ness

noun

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 T⊨ϕ is true, then T⊢ϕ must also be true, for any wff φ of logic L.
   THEOREM 37°. (Gödel's completeness theorem 1930.) In the predicate calculus H: (a) If ⊨F [א₀-⊨F], then ⊢F. If E_1,...,E_k⊨F [E_1,...,E_kא₀-⊨F], then E_1,...,E_k⊢F. (b) […] 2002, Stephen Cole Kleene, Mathematical Logic, Mineola, New York: Dover Publications, page 314