Compactness theorem

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 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;^[1] 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.

History

Kurt Gödel proved the countable compactness theorem in 1930. Anatoly Maltsev proved the uncountable case in 1936.^[2] ^[3]

Applications

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

The compactness theorem implies Robinson's principle: If a first-order sentence holds in 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 φ is a sentence that holds in every field of characteristic zero. Then its negation ¬φ, together with the field axioms and the infinite sequence of sentences 1+1 ≠ 0, 1+1+1 ≠ 0, …, is not satisfiable (because there is no field of characteristic 0 in which ¬φ holds, and the infinite sequence of sentences ensures any model would be a field of characteristic 0). Therefore, there is a finite subset A of these sentences that is not satisfiable. We can assume that A contains ¬φ, the field axioms, and, for some k, the first k sentences of the form 1+1+...+1 ≠ 0 (because adding more sentences doesn't change unsatisfiability). Let B contain all the sentences of A except ¬φ. Then any field with a characteristic greater than k is a model of B, and ¬φ together with B is not satisfiable. This means that φ must hold in every model of B, which means precisely that φ holds in every field of characteristic greater than k.

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 with uncountably many 'natural numbers'. To achieve this, let T be the initial theory and let κ be any cardinal number. Add to the language of T one constant symbol for every element of κ. Then add to T a collection of sentences that say that the objects denoted by any two distinct constant symbols from the new collection are distinct (this is a collection of κ² sentences). Since every finite subset of this new theory is satisfiable by a sufficiently large finite model of T, or by any infinite model, the entire extended theory is satisfiable. But any model of the extended theory has cardinality at least κ

A third application of the compactness theorem is the construction of nonstandard models of the real numbers, that is, consistent extensions of the theory of the real numbers that contain "infinitesimal" numbers. To see this, let Σ be a first-order axiomatization of the theory of the real numbers. Consider the theory obtained by adding a new constant symbol ε to the language and adjoining to Σ the axiom ε > 0 and the axioms ε < 1/n for all positive integers n. Clearly, the standard real numbers R are a model for every finite subset of these axioms, because the real numbers satisfy everything in Σ and, by suitable choice of ε, can be made to satisfy any finite subset of the axioms about ε. By the compactness theorem, there is a model *R that satisfies Σ and also contains an infinitesimal element ε. A similar argument, adjoining axioms ω > 0, ω > 1, etc., shows that the existence of infinitely large integers cannot be ruled out by any axiomatization Σ of the reals.^[4]

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.^[5]

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 of these sets A_i generates a proper 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's 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.

Notes

↑ See Truss (1997).
↑ Vaught, Robert L.: Alfred Tarski's work in model theory. J. Symbolic Logic 51 (1986), no. 4, 869–882
↑ Robinson, A.: Non-standard analysis. North-Holland Publishing Co., Amsterdam 1966. page 48.
↑ Goldblatt, Robert (1998). Lectures on the Hyperreals. New York: Springer. pp. 10–11. ISBN 0-387-98464-X.
↑ See Hodges (1993).

References

Boolos, George; Jeffrey, Richard; Burgess, John (2004). Computability and Logic (fourth ed.). "Cambridge University Press.
Chang, C.C.; Keisler, H. Jerome (1989). Model Theory (third ed.). Elsevier. ISBN 0-7204-0692-7.
Dawson, John W. junior (1993). "The compactness of first-order logic: From Gödel to Lindström". History and Philosophy of Logic. 14: 15–37. doi:10.1080/01445349308837208.
Hodges, Wilfrid (1993). Model theory. Cambridge University Press. ISBN 0-521-30442-3.
Marker, David (2002). Model Theory: An Introduction. Graduate Texts in Mathematics 217. Springer. ISBN 0-387-98760-6.
Truss, John K. (1997). Foundations of Mathematical Analysis. Oxford University Press. ISBN 0-19-853375-6.