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,[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, except for a very limited number of examples.[2]

History edit

Kurt Gödel proved the countable compactness theorem in 1930. Anatoly Maltsev proved the uncountable case in 1936.[3][4]

Applications edit

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

Robinson's principle edit

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

Robinson's principle:[5][6] 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 some (or equivalently, in every) 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 some algebraically closed field of characteristic   in which case   is true in all algebraically closed fields of sufficiently large non-0 characteristic  [5] One consequence is the following special case of the Ax–Grothendieck theorem: all injective complex polynomials   are surjective[5] (indeed, it can even be shown that its inverse will also be a polynomial).[7] In fact, the surjectivity conclusion remains true for any injective polynomial   where   is a finite field or the algebraic closure of such a field.[7]

Upward Löwenheim–Skolem theorem edit

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 finite 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 edit

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.[8]

It can be shown that the hyperreal numbers   satisfy the transfer principle:[9] a first-order sentence is true of   if and only if it is true of  

Proofs edit

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.[10]

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 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   and let   be a collection of  -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 edit

Notes edit

  1. ^ See Truss (1997).
  2. ^ J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985) [1], in particular, Makowsky, J. A. Chapter XVIII: Compactness, Embeddings and Definability. 645--716, see Theorems 4.5.9, 4.6.12 and Proposition 4.6.9. For compact logics for an extended notion of model see Ziegler, M. Chapter XV: Topological Model Theory. 557--577. For logics without the relativization property it is possible to have simultaneously compactness and interpolation, while the problem is still open for logics with relativization. See Xavier Caicedo, A Simple Solution to Friedman's Fourth Problem, J. Symbolic Logic, Volume 51, Issue 3 (1986), 778-784.doi:10.2307/2274031 JSTOR 2274031
  3. ^ Vaught, Robert L.: "Alfred Tarski's work in model theory". Journal of Symbolic Logic 51 (1986), no. 4, 869–882
  4. ^ Robinson, A.: Non-standard analysis. North-Holland Publishing Co., Amsterdam 1966. page 48.
  5. ^ a b c Marker 2002, pp. 40–43.
  6. ^ Gowers, Barrow-Green & Leader 2008, pp. 639–643.
  7. ^ a b Terence, Tao (7 March 2009). "Infinite fields, finite fields, and the Ax-Grothendieck theorem".
  8. ^ Goldblatt 1998, pp. 10–11.
  9. ^ Goldblatt 1998, p. 11.
  10. ^ See Hodges (1993).

References edit

External links edit