A SHORT PROOF OF PITT'S COMPACTNESS THEOREM A bounded linear operator between two Banach spaces X and Y is said to be compact
Gödel's completeness theorem - Wikipedia
Carol Karp. An algebraic proof of the Barwise compactness theorem. The syntax and semantics of infinitary languages, edited by Jon Barwise, Lecture notes in mathematics, no. 72, Springer-Verlag, Berlin, Heidelberg, and New