Compactness theorem explained
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,[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
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 (or equivalently, in) 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 algebraically closed field of characteristic
in which case
is true in 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).
[5] 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 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
\omega>0, \omega>1,\ldots,
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.[6]
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 but not to . 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
Ai=\{j\inI:j\supseteqi\}.
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
} is in
}, then
hence
holds in
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
References
- Book: Boolos, George. Jeffrey, Richard. Burgess, John. Computability and Logic. fourth. 2004. Cambridge University Press.
- Book: Chang, C.C.. Keisler, H. Jerome. Howard Jerome Keisler. Elsevier. Model Theory. 1989. third. 0-7204-0692-7.
- Dawson. John W. junior. The compactness of first-order logic: From Gödel to Lindström. History and Philosophy of Logic. 1993. 14. 15–37. 10.1080/01445349308837208.
- Book: Hodges, Wilfrid. Wilfrid Hodges. Cambridge University Press. Model theory. registration. 1993. 0-521-30442-3.
- Book: Goldblatt, Robert. Lectures on the Hyperreals. limited. Robert Goldblatt. 1998. Springer Verlag. New York. 0-387-98464-X.
- Book: Gowers. Timothy. Barrow-Green. June. Leader. Imre. The Princeton Companion to Mathematics. Princeton University Press. Princeton. 2008. 978-1-4008-3039-8. 659590835. 635–646.
- Book: Marker, David. Model Theory: An Introduction. Springer. Graduate Texts in Mathematics. 217. 2002. 978-0-387-98760-6. 49326991.
- Robinson. J. A.. A Machine-Oriented Logic Based on the Resolution Principle. Journal of the ACM. Association for Computing Machinery (ACM). 12. 1. 1965. 0004-5411. 10.1145/321250.321253. 23–41. 14389185. free.
- Book: Truss, John K.. John Truss. Foundations of Mathematical Analysis. 1997. Oxford University Press. 0-19-853375-6.
External links
Notes and References
- See Truss (1997).
- J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985) https://projecteuclid.org/euclid.pl/1235417263#toc, 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.
- [Robert Lawson Vaught|Vaught, Robert L.]
- [Abraham Robinson|Robinson, A.]
- Web site: Terence. Tao. Infinite fields, finite fields, and the Ax-Grothendieck theorem. 7 March 2009.
- See Hodges (1993).