In

Compactness Theorem

''

mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...

, the compactness theorem states that a set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...

of first-order sentences has a model
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure.
Models c ...

if and only if every finite subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...

of it has a model. This theorem is an important tool in model theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...

, 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
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations ...

is a consequence of Tychonoff's theorem (which says that the product of compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...

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 space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...

s: a collection of closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a clo ...

s in a compact space has a non-empty intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...

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
Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an im ...

proved the countable compactness theorem in 1930. Anatoly Maltsev Anatoly Ivanovich Maltsev (also: Malcev, Mal'cev; Russian: Анато́лий Ива́нович Ма́льцев; 27 November N.S./14 November O.S. 1909, Moscow Governorate – 7 June 1967, Novosibirsk) was born in Misheronsky, near Moscow, and ...

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 byAbraham Robinson
Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincor ...

in his 1949
Events
January
* January 1 – A United Nations-sponsored ceasefire brings an end to the Indo-Pakistani War of 1947. The war results in a stalemate and the division of Kashmir, which still continues as of 2022.
* January 2 – L ...

dissertation.
Robinson's principle Robinsons or Robinson's may refer to:
Businesses Department stores
* Robinsons Malls, shopping mall and retail operator in the Philippines
* Robinsons, former department store chain owned by Robinson & Co. in Singapore and Malaysia
* Robinson Dep ...

: 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 $\backslash varphi$ is a sentence that holds in every field of characteristic zero. Then its negation $\backslash lnot\; \backslash varphi,$ together with the field axioms and the infinite sequence of sentences
$$1\; +\; 1\; \backslash neq\; 0,\; \backslash ;\backslash ;\; 1\; +\; 1\; +\; 1\; \backslash neq\; 0,\; \backslash ;\; \backslash ldots$$
is not satisfiable (because there is no field of characteristic 0 in which $\backslash lnot\; \backslash varphi$ holds, and the infinite sequence of sentences ensures any model would be a field of characteristic 0). Therefore, there is a finite subset $A$ of these sentences that is not satisfiable. $A$ must contain $\backslash lnot\; \backslash varphi$ because otherwise it would be satisfiable. Because adding more sentences to $A$ does not change unsatisfiability, we can assume that $A$ contains the field axioms and, for some $k,$ the first $k$ sentences of the form $1\; +\; 1\; +\; \backslash cdots\; +\; 1\; \backslash neq\; 0.$ Let $B$ contain all the sentences of $A$ except $\backslash lnot\; \backslash varphi.$ Then any field with a characteristic greater than $k$ is a model of $B,$ and $\backslash lnot\; \backslash varphi$ together with $B$ is not satisfiable. This means that $\backslash varphi$ must hold in every model of $B,$ which means precisely that $\backslash varphi$ holds in every field of characteristic greater than $k.$ This completes the proof.
The Lefschetz principle
In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally b ...

, one of the first examples of a transfer principle
In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure. One of the first examples was the Lefschetz principle, which states that any sentence in the first ...

, extends this result. A first-order sentence $\backslash varphi$ in the language of rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...

is true in (or equivalently, in ) algebraically closed
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...

field of characteristic 0 (such as the complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

s for instance) if and only if there exist infinitely many primes $p$ for which $\backslash varphi$ is true in algebraically closed field of characteristic $p,$ in which case $\backslash varphi$ is true in algebraically closed fields of sufficiently large non-0 characteristic $p.$
One consequence is the following special case of the Ax–Grothendieck theorem: all injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...

complex polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exam ...

s $\backslash Complex^n\; \backslash to\; \backslash Complex^n$ are surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...

(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 $F^n\; \backslash to\; F^n$ where $F$ 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 largecardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...

(this is the Upward Löwenheim–Skolem theorem). So for instance, there are nonstandard models of Peano arithmetic
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly ...

with uncountably many 'natural numbers'. To achieve this, let $T$ be the initial theory and let $\backslash kappa$ be any cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. The ...

. Add to the language of $T$ one constant symbol for every element of $\backslash kappa.$ 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 $\backslash kappa^2$ sentences). Since every 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 $\backslash kappa$.
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 $\backslash Sigma$ be a first-order axiomatization of the theory of the real numbers. Consider the theory obtained by adding a new constant symbol $\backslash varepsilon$ to the language and adjoining to $\backslash Sigma$ the axiom $\backslash varepsilon\; >\; 0$ and the axioms $\backslash varepsilon\; <\; \backslash tfrac$ for all positive integers $n.$ Clearly, the standard real numbers $\backslash R$ are a model for every finite subset of these axioms, because the real numbers satisfy everything in $\backslash Sigma$ and, by suitable choice of $\backslash varepsilon,$ can be made to satisfy any finite subset of the axioms about $\backslash varepsilon.$ By the compactness theorem, there is a model $^*\; \backslash R$ that satisfies $\backslash Sigma$ and also contains an infinitesimal element $\backslash varepsilon.$ A similar argument, this time adjoining the axioms $\backslash omega\; >\; 0,\; \backslash ;\; \backslash omega\; >\; 1,\; \backslash ldots,$ etc., shows that the existence of numbers with infinitely large magnitudes cannot be ruled out by any axiomatization $\backslash Sigma$ of the reals. It can be shown that thehyperreal number
In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains number ...

s $^*\; \backslash R$ satisfy the transfer principle
In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure. One of the first examples was the Lefschetz principle, which states that any sentence in the first ...

: a first-order sentence is true of $\backslash R$ if and only if it is true of $^*\; \backslash R.$
Proofs

One can prove the compactness theorem usingGödel'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.
The completeness theorem applies to any first-order theory: ...

, 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
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection o ...

.See Hodges (1993).
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 ultraproduct
The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All fact ...

s hinging on the axiom of choice as follows:
Proof:
Fix a first-order language $L,$ and let $\backslash Sigma$ be a collection of L-sentences such that every finite subcollection of $L$-sentences, $i\; \backslash subseteq\; \backslash Sigma$ of it has a model $\backslash mathcal\_i.$ Also let $\backslash prod\_\backslash mathcal\_i$ be the direct product of the structures and $I$ be the collection of finite subsets of $\backslash Sigma.$ For each $i\; \backslash in\; I,$ let $A\_i\; =\; \backslash .$
The family of all of these sets $A\_i$ generates a proper filter
Filter, filtering or filters may refer to:
Science and technology
Computing
* Filter (higher-order function), in functional programming
* Filter (software), a computer program to process a data stream
* Filter (video), a software component tha ...

, so there is an ultrafilter $U$ containing all sets of the form $A\_i.$
Now for any formula $\backslash varphi$ in $\backslash Sigma:$
* the set $A\_$ is in $U$
* whenever $j\; \backslash in\; A\_,$ then $\backslash varphi\; \backslash in\; j,$ hence $\backslash varphi$ holds in $\backslash mathcal\; M\_j$
* the set of all $j$ with the property that $\backslash varphi$ holds in $\backslash mathcal\; M\_j$ is a superset of $A\_,$ hence also in $U$
Łoś's theorem now implies that $\backslash varphi$ holds in the ultraproduct
The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All fact ...

$\backslash prod\_\; \backslash mathcal\_i/U.$ So this ultraproduct satisfies all formulas in $\backslash Sigma.$
See also

* * * *Notes

References

* * * * * * * * *External links

Compactness Theorem

''

Internet Encyclopedia of Philosophy
The ''Internet Encyclopedia of Philosophy'' (''IEP'') is a scholarly online encyclopedia, dealing with philosophy, philosophical topics, and philosophers. The IEP combines open access publication with peer reviewed publication of original papers ...

''.
{{Mathematical logic
Mathematical logic
Metatheorems
Model theory
Theorems in the foundations of mathematics