Inmathematical logic, the compactness theorem states that a setoffirst-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 productofcompact 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]
Kurt Gödel proved the countable compactness theorem in 1930. Anatoly Maltsev proved the uncountable case in 1936.[3][4]
The compactness theorem has many applications in model theory; a few typical results are sketched here.
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 fieldofcharacteristic 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]
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
.
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
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 sentence in
Łoś's theorem now implies that holds in the ultraproduct
So this ultraproduct satisfies all formulas in
| |||||||||
---|---|---|---|---|---|---|---|---|---|
General |
| ||||||||
Theorems (list) and paradoxes |
| ||||||||
Logics |
| ||||||||
Set theory |
| ||||||||
Formal systems (list), language and syntax |
| ||||||||
Proof theory |
| ||||||||
Model theory |
| ||||||||
Computability theory |
| ||||||||
Related |
| ||||||||