Compactness Theorem - Proofs

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 proofs 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, i.e., proofs that refer to truth but not to provability. One of those proofs relies on ultraproducts hinging on the axiom of choice as follows:

Proof: Fix a first-order language L, and let Σ be a collection of L-sentences such that every finite subcollection of L-sentences, i ⊆ Σ of it has a model . Also let be the direct product of the structures and I be the collection of finite subsets of Σ. For each i in I let A_i := { j ∈ I : j ⊇ i}. The family of all these sets A_i generates a filter, so there is an ultrafilter U containing all sets of the form A_i.

Now for any formula φ in Σ we have:

• the set A_{φ} is in U
• whenever j ∈ A_{φ}, then φ ∈ j, hence φ holds in
• the set of all j with the property that φ holds in is a superset of A_{φ}, hence also in U

Using Łoś's theorem we see that φ holds in the ultraproduct . So this ultraproduct satisfies all formulas in Σ.