completenesses

Noun

completeness (usually uncountable, plural completenesses)

1. (usually) The state or condition of being complete.
2. (usually) 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 #92;vDashF [or even if #92;aleph#95;0-#92;vDashF], then #92;vdashF. If E#95;1,...,E#95;k#92;vDashF [or even if E#95;1,...,E#95;k#92;#92;aleph#95;0-#92;vDashF], then E#95;1,...,E#95;k#92;vdashF.
     (b) […]2002, Stephen Cole Kleene, Mathematical Logic, Mineola, New York: Dover Publications, →ISBN, page 314