In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful (but generally not effective) method for constructing models of any set of sentences that is finitely consistent.

The compactness theorem for the propositional calculus is a consequence of Tychonoff's theorem (which says that the product of compact spaces is compact) applied to compact Stone spaces, hence the theorem's name. Likewise, it is analogous to the finite intersection property characterization of compactness in topological spaces: a collection of closed sets in a compact space has a non-empty intersection if every finite subcollection has a non-empty intersection.

The compactness theorem is one of the two key properties, along with the downward Löwenheim–Skolem theorem, that is used in Lindström's theorem to characterize first-order logic. Although there are some generalizations of the compactness theorem to non-first-order logics, the compactness theorem itself does not hold in them, except for a very limited number of examples.

History

Kurt Gödel proved the countable compactness theorem in 1930. Anatoly Maltsev proved the uncountable case in 1936.

Applications

The compactness theorem has many applications in model theory; a few typical results are sketched here.

Robinson's principle

The compactness theorem implies the following result, stated by Abraham Robinson in his 1949 dissertation.

Robinson's principle: If a first-order sentence holds in every field of characteristic is not satisfiable This completes the proof.

The Lefschetz principle, one of the first examples of a transfer principle in the language of rings is true in (or equivalently, in ) algebraically closed field of characteristic 0 (such as the complex numbers is true in is true in One consequence is the following special case of the Ax–Grothendieck theorem: all injective complex polynomials are surjective (indeed, it can even be shown that its inverse will also be a polynomial). is a finite field or the algebraic closure of such a field.

Upward Löwenheim–Skolem theorem

A second application of the compactness theorem shows that any theory that has arbitrarily large finite models, or a single infinite model, has models of arbitrary large cardinality (this is the Upward Löwenheim–Skolem theorem). So for instance, there are nonstandard models of Peano arithmetic be any cardinal number sentences). Since every .

Non-standard analysis

A third application of the compactness theorem is the construction of nonstandard models

of the reals.

It can be shown that the hyperreal numbers satisfy the transfer principle:

Proofs

One can prove the compactness theorem using Gödel's completeness theorem, which establishes that a set of sentences is satisfiable if and only if no contradiction can be proven from it. Since proofss are always finite and therefore involve only finitely many of the given sentences, the compactness theorem follows. In fact, the compactness theorem is equivalent to Gödel's completeness theorem, and both are equivalent to the Boolean prime ideal theorem, a weak form of the axiom of choice.

Gödel originally proved the compactness theorem in just this way, but later some "purely semantic" proofs of the compactness theorem were found; that is, proofs that refer to instead of . One of those proofs relies on ultraproducts hinging on the axiom of choice as follows:

Proof generates a proper filter, so there is an ultrafilter

Łoś's theorem holds in the ultraproduct

See also

Notes

References

External links

• Compactness Theorem, Internet Encyclopedia of Philosophy.