Compactness theorem
In
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
Kurt Gödel proved the countable compactness theorem in 1930. Anatoly Maltsev proved the uncountable case in 1936.[3][4]
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:[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
The
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
Non-standard analysis
A third application of the compactness theorem is the construction of
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
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
Now for any sentence 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
- Barwise compactness theorem
- Herbrand's theorem – reduction of first-order mathematical logic to propositional logic
- List of Boolean algebra topics
- Löwenheim–Skolem theorem – Existence and cardinality of models of logical theories
Notes
- ^ See Truss (1997).
- JSTOR 2274031
- ^ Vaught, Robert L.: "Alfred Tarski's work in model theory". Journal of Symbolic Logic 51 (1986), no. 4, 869–882
- ^ Robinson, A.: Non-standard analysis. North-Holland Publishing Co., Amsterdam 1966. page 48.
- ^ a b c Marker 2002, pp. 40–43.
- ^ Gowers, Barrow-Green & Leader 2008, pp. 639–643.
- ^ a b Terence, Tao (7 March 2009). "Infinite fields, finite fields, and the Ax-Grothendieck theorem".
- ^ Goldblatt 1998, pp. 10–11.
- ^ Goldblatt 1998, p. 11.
- ^ See Hodges (1993).
References
- Boolos, George; Jeffrey, Richard; Burgess, John (2004). Computability and Logic (fourth ed.). Cambridge University Press.
- Chang, C.C.; 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. .
- ISBN 0-521-30442-3.
- ISBN 0-387-98464-X.
- Gowers, Timothy; Barrow-Green, June; Leader, Imre (2008). The Princeton Companion to Mathematics. Princeton: Princeton University Press. pp. 635–646. OCLC 659590835.
- Marker, David (2002). Model Theory: An Introduction. OCLC 49326991.
- Robinson, J. A. (1965). "A Machine-Oriented Logic Based on the Resolution Principle". Journal of the ACM. 12 (1). Association for Computing Machinery (ACM): 23–41. S2CID 14389185.
- ISBN 0-19-853375-6.