Seonglae ChoGödel's completeness theorem
The set of propositions provable in Predicate Logic has a Model Theory. In other words, the truth defined by Proof Theory and the truth defined by Model Theory coincide which means If a proposition is provable in a logical system, then that proposition is true in all models of that system.
This theorem does not contradict Gödel’s incompleteness theorem but is not satisfied in Higher-order logics.
Gödel's completeness theorem
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic.
https://en.wikipedia.org/wiki/Gödel's_completeness_theorem
builds.openlogicproject.org
https://builds.openlogicproject.org/content/first-order-logic/completeness/completeness.pdf
builds.openlogicproject.org
https://builds.openlogicproject.org/content/first-order-logic/completeness/completeness.pdf