Completeness, Compactness, and Strong Completeness
A theory is any set of statements. The truth of statements in models are defined by recursion and will agree with the definition for finitary logic where both are defined. Given a theory T a statement is said to be valid for the theory T if it is true in all models of T.
A logic is complete if for every sentence S valid in every model there exists a proof of S. It is strongly complete if for any theory T for every sentence S valid in T there is a proof of S from T. An infinitary logic can be complete without being strongly complete.
A logic is compact if for every theory T of cardinality if all subsets S of T have models then T has a model. A logic is strongly compact if for every theory T if all subsets S of T, where S has cardinality, have models then T has a model. If a logic is strongly compact, and complete, then it is strongly complete.
The cardinal is weakly compact if is compact and is strongly compact if is strongly compact.
Read more about this topic: Infinitary Logic
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