The Compactness Theorem states that if T is a collection of first-order statements and every finite subset of T is consistent, then T is itself consistent. A set of statements is consistent if it has a model.

By abuse of terminology, the following related fact is also frequently referred to as "compactness." Let M be a -saturated model, be a definable set, and be a collection of definable subsets of , with of size less than . If covers D, then some finite subset of covers D. This fact follows from the definition of -saturation. Compactness is used to prove the existence of -saturated models, however.

## Common Corollaries and Uses of Compactness

The compactness theorem has a number of commonly-used corollaries. These corollaries are often used implicitly in proofs, and explained only as "compactness."

Lemma: Let M be a model, and be a consistent partial type over M. Then there is an elementary extension in which is realized.

Proof: Apply the Compactness Theorem to the union of the elementary diagram of M and the statements , where is a new constant symbol. QED

Lemma: Let T be a theory. Let be a formula, and let be a collection of formulas. Suppose that in every model M of T, we have

.

Then there is a finite subset such that in every model M of T,

.

Proof: Apply compactness to the union of T and the statements

where is a new constant symbol. By assumption, this collection of statements is inconsistent, so by compactness, some finite subset is inconsistent. This yields . QED

Lemma: Let T be a theory, and be a formula. Suppose that in every model M of T, the set is finite. Then there is a number n such that for every model M.

Proof: Apply compactness to the union of T and the set of sentences asserting for each n that at least n elements of the model satisfy . By assumption, this is inconsistent. Consequently, there is some n such that is inconsistent with T. This means exactly that in every model of T. QED

The analogous statement in saturated models is the following: Lemma: Let M be a -saturated model. Suppose that is a formula such that for every , the set is finite. Then there is a uniform bound n on the size of .

Other important and prototypical applications of compactness include the following:

## Interpretation as Compactness of Stone Space

Let S denote the space of all complete theories (in some fixed first-order language). For each sentence , let

There is a natural topology on S in which the sets of the form are the basic open (and basic closed) subsets. The compactness theorem says exactly that S is compact with this topology.

## Proofs of Compactness

Compactness follows easily from some forms of Gödel's Completeness Theorem. Specifically, a theory is inconsistent if and only if holds in all models of . By the Completeness Theorem, this holds if and only if can be proven from . But proofs are finitary, so any proof must take only finitely many steps, and must use only a finite subset of . In particular, if proves , then so does some finite subset . So if is inconsistent, so is a finite subset.

Compactness can alternatively be proven from Łoś's Theorem.

Proof: Let T be a collection of statements, with every finite subset of T being consistent. Let X be the set of all finite of T. For each finite subset , let

Note that and that for any S. It follows that any finite intersection of 's is non-empty. Therefore we can find an ultrafilter on X such that for every S. In other words, for each S, thinks that "most" elements of X contain S.

For each finite subset S of T, we can find a model of S, by assumption. Consider the ultraproduct

We claim that M is the desired model of T. Let be a formula in T. Then

for every , or equivalently, for every . But

.

Consequently, the set of S such that is "large" with respect to the ultrafilter . By Łoś's Theorem, holds in the ultraproduct M. QED ::: ::: :::