In mathematical logic, the compactness theorem states that a set of firstorder 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 nonempty intersection if every finite subcollection has a nonempty 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 firstorder logic. Although there are some generalizations of the compactness theorem to nonfirstorder logics, the compactness theorem itself does not hold in them, except for a very limited number of examples.^{[2]}
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Compactness

Compactness Theorem and Expressive Limitations of First Order Logic

Sequential Compactness

Compactness theorems / Topology / Maths

What Is COMPACTNESS THEOREM? COMPACTNESS THEOREM Definition & Meaning
Transcription
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 firstorder 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 Lefschetz principle, one of the first examples of a transfer principle, extends this result. A firstorder 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 non0 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
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 .
Nonstandard 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 firstorder 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 firstorder 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 firstorder language and let be a collection of Lsentences 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
 Barwise compactness theorem
 Herbrand's theorem – reduction of firstorder 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).
 ^ J. Barwise, S. Feferman, eds., ModelTheoretic Logics (New York: SpringerVerlag, 1985) [1], in particular, Makowsky, J. A. Chapter XVIII: Compactness, Embeddings and Definability. 645716, 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. 557577. 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), 778784.doi:10.2307/2274031 JSTOR 2274031
 ^ Vaught, Robert L.: "Alfred Tarski's work in model theory". Journal of Symbolic Logic 51 (1986), no. 4, 869–882
 ^ Robinson, A.: Nonstandard analysis. NorthHolland Publishing Co., Amsterdam 1966. page 48.
 ^ ^{a} ^{b} ^{c} Marker 2002, pp. 40–43.
 ^ Gowers, BarrowGreen & Leader 2008, pp. 639–643.
 ^ ^{a} ^{b} Terence, Tao (7 March 2009). "Infinite fields, finite fields, and the AxGrothendieck 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.; Keisler, H. Jerome (1989). Model Theory (third ed.). Elsevier. ISBN 0720406927.
 Dawson, John W. junior (1993). "The compactness of firstorder 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 0521304423.
 Goldblatt, Robert (1998). Lectures on the Hyperreals. New York: Springer Verlag. ISBN 038798464X.
 Gowers, Timothy; BarrowGreen, June; Leader, Imre (2008). The Princeton Companion to Mathematics. Princeton: Princeton University Press. pp. 635–646. ISBN 9781400830398. OCLC 659590835.
 Marker, David (2002). Model Theory: An Introduction. Graduate Texts in Mathematics. Vol. 217. Springer. ISBN 9780387987606. OCLC 49326991.
 Robinson, J. A. (1965). "A MachineOriented Logic Based on the Resolution Principle". Journal of the ACM. Association for Computing Machinery (ACM). 12 (1): 23–41. doi:10.1145/321250.321253. ISSN 00045411. S2CID 14389185.
 Truss, John K. (1997). Foundations of Mathematical Analysis. Oxford University Press. ISBN 0198533756.