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.
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. 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.
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 <math>\Sigma</math> be a collection of L-sentences such that every finite subcollection of L-sentences, <math>i \subseteq \Sigma</math> of it has a model <math>\mathcal{M}_i</math>. Also let <math>\prod_{i \subseteq \Sigma}\mathcal{M}_i</math> be the direct product of the structures and <math>I</math> be the collection of nonempty finite subsets of <math>\Sigma</math>. Then fix an arbitrary ultrafilter, <math>\mathcal{U}</math> on <math>I</math> consisting of all finitely generated maximal chains in <math>I</math> (i.e. all i* = <math>\{j| j \in I; j \supseteq i\}</math> for i a nonempty finite subset of <math>\Sigma</math>). Then for any <math>\varphi \in \Sigma</math>, <math>\mathcal{M}_i \models \varphi</math> for all <math>i \in \{\varphi\}^*</math> so that the indicies, <math>i \in I</math> for which <math>\varphi</math> holds in <math>\mathcal{M}_i</math>, <math>\Vert \varphi \Vert \supseteq \{\varphi\}^* \in \mathcal{U}</math>. Then by the closure of supersets in ultrafilters, we have <math>\Vert \varphi \Vert \in \mathcal{U}</math> so that we get that the ultraproduct <math>\prod_{i \subseteq \Sigma}\mathcal{M}_i/\mathcal{U} \models \varphi</math> by Łoś's Theorem. (Rothmaler 2000)
See also
it:Teorema di compattezza (semantico) he:קומפקטיות pl:Twierdzenie o zwartości