completeness

noun

Etymology: Etymology tree English complete Proto-Germanic *-in- Proto-Indo-European *-h₂ Proto-Indo-European *-éh₂ Proto-Indo-European *-yéti Proto-Indo-European *-eh₂yéti Proto-Indo-European *-h₂ti Proto-Germanic *-ōną Proto-Germanic *-inōną Proto-Indo-European *-dyé- Proto-Germanic *-atjaną Proto-Indo-European *-tus Proto-Germanic *-þuz Proto-Germanic *-assuz Proto-Germanic *-inassuz Proto-West Germanic *-nassī Old English -nes Middle English -nesse English -ness English completeness From complete + -ness.

1. The state or condition of being complete.
2. 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) […]”