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 (but generally not effective) 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, 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, except for a very limited number of examples.

History
Kurt Gödel proved the countable compactness theorem in 1930.  Anatoly Maltsev proved the uncountable case in 1936.

Applications

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

Robinson's principle

The compactness theorem implies the following result, stated by Abraham Robinson in his 1949 dissertation. 

Robinson's principle: If a first-order sentence holds in every field of characteristic zero, then there exists a constant  such that the sentence holds for every field of characteristic larger than  This can be seen as follows: suppose  is a sentence that holds in every field of characteristic zero. Then its negation  together with the field axioms and the infinite sequence of sentences 

is not satisfiable (because there is no field of characteristic 0 in which  holds, and the infinite sequence of sentences ensures any model would be a field of characteristic 0). Therefore, there is a finite subset  of these sentences that is not satisfiable.  must contain  because otherwise it would be satisfiable. Because adding more sentences to  does not change unsatisfiability, we can assume that  contains the field axioms and, for some  the first  sentences of the form  Let  contain all the sentences of  except  Then any field with a characteristic greater than  is a model of  and  together with  is not satisfiable. This means that  must hold in every model of  which means precisely that  holds in every field of characteristic greater than  This completes the proof. 

The Lefschetz principle, one of the first examples of a transfer principle, extends this result. A first-order sentence  in the language of rings is true in  (or equivalently, in ) algebraically closed field of characteristic 0 (such as the complex numbers for instance) if and only if there exist infinitely many primes  for which  is true in  algebraically closed field of characteristic  in which case  is true in  algebraically closed fields of sufficiently large non-0 characteristic  
One consequence is the following special case of the Ax–Grothendieck theorem: all injective complex polynomials  are surjective (indeed, it can even be shown that its inverse will also be a polynomial). In fact, the surjectivity conclusion remains true for any injective polynomial  where  is a finite field or the algebraic closure of such a field.

Upward Löwenheim–Skolem theorem

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  be the initial theory and let  be any cardinal number. Add to the language of  one constant symbol for every element of  Then add to  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  sentences). Since every  subset of this new theory is satisfiable by a sufficiently large finite model of  or by any infinite model, the entire extended theory is satisfiable. But any model of the extended theory has cardinality at least .

Non-standard analysis

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  be a first-order axiomatization of the theory of the real numbers.  Consider the theory obtained by adding a new constant symbol  to the language and adjoining to  the axiom  and the axioms  for all positive integers   Clearly, the standard real numbers  are a model for every finite subset of these axioms, because the real numbers satisfy everything in  and, by suitable choice of  can be made to satisfy any finite subset of the axioms about   By the compactness theorem, there is a model  that satisfies  and also contains an infinitesimal element   

A similar argument, this time adjoining the axioms  etc., shows that the existence of numbers with infinitely large magnitudes cannot be ruled out by any axiomatization  of the reals. 

It can be shown that the hyperreal numbers  satisfy the transfer principle: a first-order sentence is true of  if and only if it is true of

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; that is, proofs that refer to  but not to .  One of those proofs relies on ultraproducts hinging on the axiom of choice as follows:

Proof: 
Fix a first-order language  and let   be a collection of L-sentences such that every finite subcollection of -sentences,  of it has a model   Also let  be the direct product of the structures and  be the collection of finite subsets of  For each  let  
The family of all of these sets  generates a proper filter, so there is an ultrafilter  containing all sets of the form 

Now for any formula  in 
 the set  is in 
 whenever  then  hence  holds in 
 the set of all  with the property that  holds in   is a superset of  hence also in 
Łoś's theorem now implies that  holds in the ultraproduct   So this ultraproduct satisfies all formulas in

See also

Notes

References

External links
 Compactness Theorem, Internet Encyclopedia of Philosophy.

Mathematical logic
Metatheorems
Model theory
Theorems in the foundations of mathematics