completeness

complete +‎ -ness

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 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) [...]

• (state of being complete): completion, fulfillment; see also Thesaurus:completion

• incompleteness, unfinishedness; see also Thesaurus:incompletion