completeness (uncountable)

  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 \vDash \phi is true, then T \vdash \phi 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 \vDash F [or even if \aleph_0-\vDash F], then \vdash F. If E_1, . . ., E_k \vDash F [or even if E_1, . . ., E_k \ \aleph_0-\vDash F], then E_1, . . ., E_k \vdash F.
      (b) [...]
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