Compactness theorem

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The compactness theorem is a basic fact in symbolic logic and model theory and asserts that a set (possibly infinite) of first-order sentences is satisfiable, i.e., has a model, if and only if every finite subset of it is satisfiable.

The compactness theorem for the propositional calculus is a result of Tychonoff's theorem (which says that the product of compact spaces is compact) applied to compact Stone spaces; hence the theorem's name.

1 Applications
2 Proofs
3 See also
4 References

[edit] Applications

From the theorem it follows for instance that if some first-order sentence holds for every field of characteristic zero, then there exists a constant p such that the sentence holds for every field of characteristic larger than p. This can be seen as follows: suppose S is the sentence under consideration. Then its negation ~S, together with the field axioms and the infinite series of sentences 1+1 ≠ 0, 1+1+1 ≠ 0, ... is not satisfiable by assumption. Therefore a finite subset of these sentences is not satisfiable, meaning that S holds in those fields which have large enough characteristic.

Also, it follows from the theorem that any theory that has an 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 with uncountably many 'natural numbers'. The nonstandard analysis is another example where infinite natural numbers appear, a possibility that cannot be excluded by any axiomatization - also a consequence of the compactness theorem.

[edit] 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. In fact, the compactness theorem is equivalent to Gödel's completeness theorem, and both are equivalent to the ultrafilter lemma, a weak form of the axiom of choice. Since proofs are always finite and therefore involve only finitely many of the given sentences, the compactness theorem follows.

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 $\mathcal{M}_i$ . Also let $\prod_{i \subseteq \Sigma}\mathcal{M}_i$ 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 $\mathcal M_j$
the set of all j with the property that φ holds in $\mathcal M_j$ is a superset of A_{φ}, hence also in U

Using Łoś' theorem we see that φ holds in the ultraproduct $\prod_{i \subseteq \Sigma}\mathcal{M}_i/U$ . So this ultraproduct satisfies all formulas in Σ.

[edit] See also

[edit] References

C. C. Chang, H. J. Keisler Model theory (1977) ISBN 0-7204-0692-7
David Marker Model Theory: An Introduction (2002) Springer-Verlag, ISBN 0-387-98760-6

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Categories: Model theory | Mathematical theorems