That version of the theorem says that a theory is satisfiable if and only if every finite subset of that theory is satisfiable, and can be proven using Tychonoff's theorem.
First order logic is the strongest logic satisfying both the Lowenheim-Skolem theorem and the theorem named for this property, according to Lindstrom's theorem. A simple case of that theorem for propositional logic can be proven by showing that Stone spaces have this property. That version of the theorem says that a theory is satisfiable if and only if every finite subset of that theory is satisfiable, and can be proven using Tychonoff's theorem. A real valued function on this type of space attains a maximum, according to the extreme value theorem. The sequential version of this property holds for a set in which all sequences have a convergent subsequence, which in R n is equivalent, by the Bolzano-Weierstrass theorem, to being closed and bounded. For ten points, the Heine-Borel theorem gives a characterization of this property in R n which says that any open cover has a finite subcover.
First order logic is the strongest logic satisfying both the Lowenheim-Skolem theorem and the theorem named for this property, according to Lindstrom's theorem.
{compactness}