Compactness theorem: Difference between revisions

Revision as of 02:16, 1 December 2002

Revision as of 02:17, 1 December 2002

Line 1:

In [[symbolic logic]], the '''compactness theorem''' asserts that a set of [[first-order predicate calculus|first-order]] sentences is satifiable, i.e., has a [[model theory|model]], if and only if every finite [[subset]] of it is satifiable. [[GoedelGödel]]'s [[completeness theorem]] says that any consistent set of sentences in logic has a model, so the compactness theorem is a corollary of the completeness theorem. Goedel originally proved the compactness theorem in just that way, but later some "purely semantic" proofs of the compactness theorem were found, i.e., proofs that refer to ''truth'' but not to ''provability''. The concept of "consistency", which occurs in the statement of the completeness theorem, relies essentially on the idea of "provability", since a set of sentences is consistent if and only if no contradiction is ''provable'' from sentences in the set. The ''finite'' nature of ''proofs'' entails the finiteness in the compactness theorem when Goedel's way of proving the compactness theorem is followed.

This is a basic fact in logic and [[model theory]], and has very far-reaching consequences. For instance, it follows that if some first-order sentence can be true in [[field]]s of arbitrary large [[characteristic]], it must also be true in some field of [[characteristic]] zero. In other words, if some sentence holds for every field of characteristic zero it must hold for every field of [[characteristic]] larger than some constant.