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In
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 by
Abraham 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 $\varphi$ is a sentence that holds in every field of characteristic zero. Then its negation $\lnot \varphi,$ together with the field axioms and the infinite sequence of sentences $1 + 1 \neq 0, \;\; 1 + 1 + 1 \neq 0, \; \ldots$ is not satisfiable (because there is no field of characteristic 0 in which $\lnot \varphi$ holds, and the infinite sequence of sentences ensures any model would be a field of characteristic 0). Therefore, there is a finite subset $A$ of these sentences that is not satisfiable. $A$ must contain $\lnot \varphi$ 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 $1 + 1 + \cdots + 1 \neq 0.$ Let $B$ contain all the sentences of $A$ except $\lnot \varphi.$ Then any field with a characteristic greater than $k$ is a model of $B,$ and $\lnot \varphi$ together with $B$ is not satisfiable. This means that $\varphi$ must hold in every model of $B,$ which means precisely that $\varphi$ 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 $\varphi$ 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 $\varphi$ is true in algebraically closed field of characteristic $p,$ in which case $\varphi$ 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 $\Complex^n \to \Complex^n$ 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 $F^n \to F^n$ 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 large
cardinality 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 $\kappa$ 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 $\kappa.$ 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 $\kappa^2$ 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 $\kappa$.

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 $\Sigma$ be a first-order axiomatization of the theory of the real numbers. Consider the theory obtained by adding a new constant symbol $\varepsilon$ to the language and adjoining to $\Sigma$ the axiom $\varepsilon > 0$ and the axioms $\varepsilon < \tfrac$ 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 $\Sigma$ and, by suitable choice of $\varepsilon,$ can be made to satisfy any finite subset of the axioms about $\varepsilon.$ By the compactness theorem, there is a model $^* \R$ that satisfies $\Sigma$ and also contains an infinitesimal element $\varepsilon.$ 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 $\Sigma$ of the reals. It can be shown that the
hyperreal 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 $^* \R$ 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 $\R$ if and only if it is true of $^* \R.$

Proofs

One can prove the compactness theorem using
Gödel'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 $\Sigma$ be a collection of L-sentences such that every finite subcollection of $L$-sentences, $i \subseteq \Sigma$ of it has a model $\mathcal_i.$ Also let $\prod_\mathcal_i$ be the direct product of the structures and $I$ be the collection of finite subsets of $\Sigma.$ For each $i \in I,$ let $A_i = \.$ 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 $\varphi$ in $\Sigma:$ * the set $A_$ is in $U$ * whenever $j \in A_,$ then $\varphi \in j,$ hence $\varphi$ holds in $\mathcal M_j$ * the set of all $j$ with the property that $\varphi$ holds in $\mathcal M_j$ is a superset of $A_,$ hence also in $U$ Łoś's theorem now implies that $\varphi$ 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 ...
$\prod_ \mathcal_i/U.$ So this ultraproduct satisfies all formulas in $\Sigma.$

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References

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