THEOREM 37°. (Gödel's completeness theorem 1930.) In the predicate calculus H:(a) If ⊨ F {\displaystyle \vDash F} [or even if ℵ 0 {\displaystyle \aleph _{0}} - ⊨ F {\displaystyle \vDash F} ], then ⊢ F {\displaystyle \vdash F} . If E 1 , . . , E k ⊨ F {\displaystyle E_{1},...,E_{k}\vDash F} [or even if E 1 , . . , E k ℵ 0 {\displaystyle E_{1},...,E_{k}\ \aleph _{0}} - ⊨ F {\displaystyle \vDash F} ], then E 1 , . . , E k ⊢ F {\displaystyle E_{1},...,E_{k}\vdash F} .(b) […]
2002, Stephen Cole Kleene, Mathematical Logic, Mineola, New York: Dover Publications, page 314