completeness

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completeness

Wikipedia

• completenesse (obsolete)

From 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, 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

• completeness axiom