If 
is a set of axioms in a first-order language, and a statement
holds for any structure 
satisfying 
, then 
can be formally deduced from 
in some appropriately defined fashion.
Gödel's Completeness Theorem
See also
Gödel's First Incompleteness Theorem, Gödel's Second Incompleteness Theorem, Löwenheim-Skolem TheoremExplore with Wolfram|Alpha
References
Beth, E. W. The Foundations of Mathematics: A Study in the Philosophy of Science. Amsterdam, Netherlands: North-Holland, 1959.Gödel, K. Über die Vollständigkeit des Logikkalküls. Doctoral dissertation. Vienna, Austria: University of Vienna, 1929.Gödel, K. `Die Vollständigkeit der Axiome des logischen Funktionenkalküls." Monatshefte für Math. u. Phys. 37, 349-360, 1930.Cite this as:
Weisstein, Eric W. "Gödel's Completeness Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GoedelsCompletenessTheorem.html