Compactness theorem
Encyclopedia
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 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 space
s 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.
The compactness theorem implies Robinson's principle: If a first-order sentence holds in every field
of characteristic
zero, then there exists a constant p such that the sentence holds for every field of characteristic larger than p. This can be seen as follows: suppose φ is a sentence that holds in every field of characteristic zero. Then its negation ¬φ, together with the field axioms and the infinite sequence of sentences 1+1 ≠ 0, 1+1+1 ≠ 0, … , is not satisfiable (because there is no field of characteristic 0 in which ¬φ holds, and the infinite sequence of sentences ensures any model would be a field of characteristic 0). Therefore, there is a finite subset A of these sentences that is not satisfiable. We can assume that A contains ¬φ, the field axioms, and, for some k, the first k sentences of the form 1+1+...+1 ≠ 0 (because adding more sentences doesn't change unsatisfiability). Let B contains all the sentences of A except ¬φ. Then any model of B is a field of characteristic greater than k, and ¬φ together with B is not satisfiable. This means that φ must hold in every model of B, which means precisely that φ holds in every field of characteristic greater than k.
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 T be the initial theory and let κ be any cardinal number
. Add to the language of T one constant symbol for every element of κ. Then add to 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 κ2 sentences). Since every finite subset of this new theory is satisfiable by a sufficiently large finite model of T, or by any infinite model, the entire extended theory is satisfiable. But any model of the extended theory has cardinality at least κ
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 Σ be a first-order axiomatization of the theory of the real numbers. Consider the theory obtained by adding a new constant symbol ε to the language and adjoining to Σ the axiom ε > 0 and the axioms ε < 1/n for all positive integers n. Clearly, the standard real numbers R are a model for every finite subset of these axioms, because the real numbers satisfy everything in Σ and, by suitable choice of ε, can be made to satisfy any finite subset of the axioms about ε. By the compactness theorem, there is a model *R that satisfies Σ and also contains an infinitesimal element ε. A similar argument, adjoining axioms ω > 0, ω > 1, etc., shows that the existence of infinitely large integers cannot be ruled out by any axiomatization Σ of the reals.
, 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, i.e., proofs that refer to truth but not to provability. One of those proofs relies on ultraproduct
s hinging on the axiom of choice as follows:
Proof: Fix a first-order language L, and let Σ be a collection of L-sentences such that every finite subcollection of L-sentences, i ⊆ Σ of it has a model
. Also let 
be the direct product of the structures and I be the collection of finite subsets of Σ. For each i in I let
Ai := { j ∈ I : j ⊇ i}.
The family of all these sets Ai generates a filter
, so there is an ultrafilter
U containing all sets of the form Ai.
Now for any formula φ in Σ we have:
Using Łoś's theorem we see that φ holds in the ultraproduct
. So this ultraproduct satisfies all formulas in Σ.
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, the compactness theorem states that a set of first-order sentences has a model
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
if and only if every finite subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of it has a model. This theorem is an important tool in model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
, as it provides a useful method for constructing models of any set of sentences that is finitely consistent
Consistency
Consistency can refer to:* Consistency , the psychological need to be consistent with prior acts and statements* "Consistency", an 1887 speech by Mark Twain...
.
The compactness theorem for the propositional calculus
Propositional calculus
In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true...
is a consequence of Tychonoff's theorem
Tychonoff's theorem
In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact. The theorem is named after Andrey Nikolayevich Tychonoff, who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the...
(which says that the product of compact space
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
s is compact) applied to compact Stone spaces; hence, the theorem's name. Likewise, it is analogous to the finite intersection property
Finite intersection property
In general topology, a branch of mathematics, the finite intersection property is a property of a collection of subsets of a set X. A collection has this property if the intersection over any finite subcollection of the collection is nonempty....
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
Löwenheim–Skolem theorem
In mathematical logic, the Löwenheim–Skolem theorem, named for Leopold Löwenheim and Thoralf Skolem, states that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ...
, that is used in Lindström's theorem
Lindström's theorem
In mathematical logic, Lindström's theorem states that first-order logic is the strongest logic In mathematical logic, Lindström's theorem (named after Swedish logician Per Lindström) states that first-order logic is the strongest logic In mathematical logic, Lindström's theorem (named after...
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.
Applications
The compactness theorem has many applications in model theory; a few typical results are sketched here.The compactness theorem implies Robinson's principle: If a first-order sentence holds in every field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
of characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
zero, then there exists a constant p such that the sentence holds for every field of characteristic larger than p. This can be seen as follows: suppose φ is a sentence that holds in every field of characteristic zero. Then its negation ¬φ, together with the field axioms and the infinite sequence of sentences 1+1 ≠ 0, 1+1+1 ≠ 0, … , is not satisfiable (because there is no field of characteristic 0 in which ¬φ holds, and the infinite sequence of sentences ensures any model would be a field of characteristic 0). Therefore, there is a finite subset A of these sentences that is not satisfiable. We can assume that A contains ¬φ, the field axioms, and, for some k, the first k sentences of the form 1+1+...+1 ≠ 0 (because adding more sentences doesn't change unsatisfiability). Let B contains all the sentences of A except ¬φ. Then any model of B is a field of characteristic greater than k, and ¬φ together with B is not satisfiable. This means that φ must hold in every model of B, which means precisely that φ holds in every field of characteristic greater than k.
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 T be the initial theory and let κ be any cardinal number
Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
. Add to the language of T one constant symbol for every element of κ. Then add to 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 κ2 sentences). Since every finite subset of this new theory is satisfiable by a sufficiently large finite model of T, or by any infinite model, the entire extended theory is satisfiable. But any model of the extended theory has cardinality at least κ
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 Σ be a first-order axiomatization of the theory of the real numbers. Consider the theory obtained by adding a new constant symbol ε to the language and adjoining to Σ the axiom ε > 0 and the axioms ε < 1/n for all positive integers n. Clearly, the standard real numbers R are a model for every finite subset of these axioms, because the real numbers satisfy everything in Σ and, by suitable choice of ε, can be made to satisfy any finite subset of the axioms about ε. By the compactness theorem, there is a model *R that satisfies Σ and also contains an infinitesimal element ε. A similar argument, adjoining axioms ω > 0, ω > 1, etc., shows that the existence of infinitely large integers cannot be ruled out by any axiomatization Σ of the reals.
Proofs
One can prove the compactness theorem using Gödel's completeness theoremGödel's completeness theorem
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It was first proved by Kurt Gödel in 1929....
, 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
Boolean prime ideal theorem
In mathematics, a prime ideal theorem guarantees the existence of certain types of subsets in a given abstract algebra. A common example is the Boolean prime ideal theorem, which states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on...
, 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, i.e., proofs that refer to truth but not to provability. One of those proofs relies on ultraproduct
Ultraproduct
The ultraproduct is a mathematical construction that appears mainly in abstract algebra and in model theory, a branch of mathematical logic. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature...
s hinging on the axiom of choice as follows:
Proof: Fix a first-order language L, and let Σ be a collection of L-sentences such that every finite subcollection of L-sentences, i ⊆ Σ of it has a model
. Also let be the direct product of the structures and I be the collection of finite subsets of Σ. For each i in I letAi := { j ∈ I : j ⊇ i}.
The family of all these sets Ai generates a filter
Filter (mathematics)
In mathematics, a filter is a special subset of a partially ordered set. A frequently used special case is the situation that the ordered set under consideration is just the power set of some set, ordered by set inclusion. Filters appear in order and lattice theory, but can also be found in...
, so there is an ultrafilter
Ultrafilter
In the mathematical field of set theory, an ultrafilter on a set X is a collection of subsets of X that is a filter, that cannot be enlarged . An ultrafilter may be considered as a finitely additive measure. Then every subset of X is either considered "almost everything" or "almost nothing"...
U containing all sets of the form Ai.
Now for any formula φ in Σ we have:
- the set A{φ} is in U
- whenever j ∈ A{φ}, then φ ∈ j, hence φ holds in
- the set of all j with the property that φ holds in
is a superset of A{φ}, hence also in U
Using Łoś's theorem we see that φ holds in the ultraproduct
Ultraproduct
The ultraproduct is a mathematical construction that appears mainly in abstract algebra and in model theory, a branch of mathematical logic. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature...
. So this ultraproduct satisfies all formulas in Σ.See also
- List of Boolean algebra topics
- Löwenheim-Skolem theorem
- Herbrand's theorem
- Barwise compactness theoremBarwise compactness theoremIn mathematical logic, the Barwise compactness theorem, named after Jon Barwise, is a generalization of the usual compactness theorem for first-order logic to a certain class of infinitary languages. It was stated and proved by Barwise in 1967....