
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.
: 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
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 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
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 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 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
- Truss 1997.
- 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
- 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.
- Marker 2002, pp. 40–43.
- Gowers, Barrow-Green & Leader 2008, pp. 639–643.
- 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.; Keisler, H. Jerome (1989). Model Theory (third ed.). Elsevier. 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. doi:10.1080/01445349308837208.
- Hodges, Wilfrid (1993). Model theory. Cambridge University Press. ISBN 0-521-30442-3.
- Goldblatt, Robert (1998). Lectures on the Hyperreals. New York: Springer Verlag. 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. ISBN 978-1-4008-3039-8. OCLC 659590835.
- Marker, David (2002). Model Theory: An Introduction. Graduate Texts in Mathematics. Vol. 217. Springer. ISBN 978-0-387-98760-6. 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. doi:10.1145/321250.321253. ISSN 0004-5411. S2CID 14389185.
- Truss, John K. (1997). Foundations of Mathematical Analysis. Oxford University Press. ISBN 0-19-853375-6.
External links
- Compactness Theorem, Internet Encyclopedia of Philosophy.
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 Lowenheim Skolem theorem that is used in Lindstrom 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 HistoryKurt Godel proved the countable compactness theorem in 1930 Anatoly Maltsev proved the uncountable case in 1936 ApplicationsThe 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 If a first order sentence holds in every field of characteristic zero then there exists a constant p displaystyle p such that the sentence holds for every field of characteristic larger than p displaystyle p This can be seen as follows suppose f displaystyle varphi is a sentence that holds in every field of characteristic zero Then its negation f displaystyle lnot varphi together with the field axioms and the infinite sequence of sentences 1 1 0 1 1 1 0 displaystyle 1 1 neq 0 1 1 1 neq 0 ldots is not satisfiable because there is no field of characteristic 0 in which f displaystyle lnot varphi holds and the infinite sequence of sentences ensures any model would be a field of characteristic 0 Therefore there is a finite subset A displaystyle A of these sentences that is not satisfiable A displaystyle A must contain f displaystyle lnot varphi because otherwise it would be satisfiable Because adding more sentences to A displaystyle A does not change unsatisfiability we can assume that A displaystyle A contains the field axioms and for some k displaystyle k the first k displaystyle k sentences of the form 1 1 1 0 displaystyle 1 1 cdots 1 neq 0 Let B displaystyle B contain all the sentences of A displaystyle A except f displaystyle lnot varphi Then any field with a characteristic greater than k displaystyle k is a model of B displaystyle B and f displaystyle lnot varphi together with B displaystyle B is not satisfiable This means that f displaystyle varphi must hold in every model of B displaystyle B which means precisely that f displaystyle varphi holds in every field of characteristic greater than k displaystyle k This completes the proof The Lefschetz principle one of the first examples of a transfer principle extends this result A first order sentence f displaystyle varphi 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 p displaystyle p for which f displaystyle varphi is true in some algebraically closed field of characteristic p displaystyle p in which case f displaystyle varphi is true in all algebraically closed fields of sufficiently large non 0 characteristic p displaystyle p One consequence is the following special case of the Ax Grothendieck theorem all injective complex polynomials Cn Cn displaystyle mathbb C n to mathbb C n 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 Fn Fn displaystyle F n to F n where F displaystyle F is a finite field or the algebraic closure of such a field Upward Lowenheim 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 Lowenheim Skolem theorem So for instance there are nonstandard models of Peano arithmetic with uncountably many natural numbers To achieve this let T displaystyle T be the initial theory and let k displaystyle kappa be any cardinal number Add to the language of T displaystyle T one constant symbol for every element of k displaystyle kappa Then add to T displaystyle T 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 k2 displaystyle kappa 2 sentences Since every finite subset of this new theory is satisfiable by a sufficiently large finite model of T displaystyle T or by any infinite model the entire extended theory is satisfiable But any model of the extended theory has cardinality at least k displaystyle kappa 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 S displaystyle Sigma be a first order axiomatization of the theory of the real numbers Consider the theory obtained by adding a new constant symbol e displaystyle varepsilon to the language and adjoining to S displaystyle Sigma the axiom e gt 0 displaystyle varepsilon gt 0 and the axioms e lt 1n displaystyle varepsilon lt tfrac 1 n for all positive integers n displaystyle n Clearly the standard real numbers R displaystyle mathbb R are a model for every finite subset of these axioms because the real numbers satisfy everything in S displaystyle Sigma and by suitable choice of e displaystyle varepsilon can be made to satisfy any finite subset of the axioms about e displaystyle varepsilon By the compactness theorem there is a model R displaystyle mathbb R that satisfies S displaystyle Sigma and also contains an infinitesimal element e displaystyle varepsilon A similar argument this time adjoining the axioms w gt 0 w gt 1 displaystyle omega gt 0 omega gt 1 ldots etc shows that the existence of numbers with infinitely large magnitudes cannot be ruled out by any axiomatization S displaystyle Sigma of the reals It can be shown that the hyperreal numbers R displaystyle mathbb R satisfy the transfer principle a first order sentence is true of R displaystyle mathbb R if and only if it is true of R displaystyle mathbb R ProofsOne can prove the compactness theorem using Godel 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 Godel s completeness theorem and both are equivalent to the Boolean prime ideal theorem a weak form of the axiom of choice Godel 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 L displaystyle L and let S displaystyle Sigma be a collection of L displaystyle L sentences such that every finite subcollection of L displaystyle L sentences i S displaystyle i subseteq Sigma of it has a model Mi displaystyle mathcal M i Also let i SMi textstyle prod i subseteq Sigma mathcal M i be the direct product of the structures and I displaystyle I be the collection of finite subsets of S displaystyle Sigma For each i I displaystyle i in I let Ai j I j i displaystyle A i j in I j supseteq i The family of all of these sets Ai displaystyle A i generates a proper filter so there is an ultrafilter U displaystyle U containing all sets of the form Ai displaystyle A i Now for any sentence f displaystyle varphi in S displaystyle Sigma the set A f displaystyle A varphi is in U displaystyle U whenever j A f displaystyle j in A varphi then f j displaystyle varphi in j hence f displaystyle varphi holds in Mj displaystyle mathcal M j the set of all j displaystyle j with the property that f displaystyle varphi holds in Mj displaystyle mathcal M j is a superset of A f displaystyle A varphi hence also in U displaystyle U Los s theorem now implies that f displaystyle varphi holds in the ultraproduct i SMi U textstyle prod i subseteq Sigma mathcal M i U So this ultraproduct satisfies all formulas in S displaystyle Sigma See alsoBarwise compactness theorem Herbrand s theorem reduction of first order mathematical logic to propositional logicPages displaying wikidata descriptions as a fallback List of Boolean algebra topics Lowenheim Skolem theorem Existence and cardinality of models of logical theoriesNotesTruss 1997 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 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 Marker 2002 pp 40 43 Gowers Barrow Green amp Leader 2008 pp 639 643 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 ReferencesBoolos 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 0 7204 0692 7 Dawson John W junior 1993 The compactness of first order logic From Godel to Lindstrom History and Philosophy of Logic 14 15 37 doi 10 1080 01445349308837208 Hodges Wilfrid 1993 Model theory Cambridge University Press ISBN 0 521 30442 3 Goldblatt Robert 1998 Lectures on the Hyperreals New York Springer Verlag 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 ISBN 978 1 4008 3039 8 OCLC 659590835 Marker David 2002 Model Theory An Introduction Graduate Texts in Mathematics Vol 217 Springer ISBN 978 0 387 98760 6 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 doi 10 1145 321250 321253 ISSN 0004 5411 S2CID 14389185 Truss John K 1997 Foundations of Mathematical Analysis Oxford University Press ISBN 0 19 853375 6 External linksCompactness Theorem Internet Encyclopedia of Philosophy