Section 1.7: Problem 3 Solution
Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
Show that from the corollary to the compactness theorem we can prove the compactness theorem itself (far more easily than we can starting from scratch).
Compactness Theorem. A set of wffs is satisfiable iff every finite subset is satisfiable.
Corollary. If , then there is a finite subset s.t. .
We prove the theorem assuming that the corollary is true. Let be a set of wffs. If is satisfiable, so is every its (finite) subset. The other direction. Suppose that every finite subset of is satisfiable. A set of wffs is unsatisfiable iff (or any other tautologically false wff). If were unsatisfiable, then , and, according to the corollary, there would be a finite subset such that , meaning that is not satisfiable, and contradicting the assumption. So, is satisfiable.