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 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.

Kurt Gödel proved the countable compactness theorem in 1930. Anatoly Maltsev proved the uncountable case in 1936.^{ [2] }^{ [3] }

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 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* contain all the sentences of *A* except ¬φ. Then any field with a characteristic greater than *k* is a model of *B*, 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 κ^{2} 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.^{ [4] }

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.^{ [5] }

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 ultraproducts 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 A_{i} := { *j* ∈ *I* : *j* ⊇ *i*}. The family of all of these sets A_{i} generates a proper filter, so there is an ultrafilter *U* containing all sets of the form A_{i}.

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 . So this ultraproduct satisfies all formulas in Σ.

- ↑ See Truss (1997).
- ↑ Vaught, Robert L.: "Alfred Tarski's work in model theory".
*Journal of Symbolic Logic*51 (1986), no. 4, 869–882 - ↑ Robinson, A.:
*Non-standard analysis*. North-Holland Publishing Co., Amsterdam 1966. page 48. - ↑ Goldblatt, Robert (1998).
*Lectures on the Hyperreals*. New York: Springer Verlag. pp. 10–11. ISBN 0-387-98464-X. - ↑ See Hodges (1993).

**First-order logic**—also known as **predicate logic**, **quantificational logic**, and **first-order predicate calculus**—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects and allows the use of sentences that contain variables, so that rather than propositions such as *Socrates is a man* one can have expressions in the form "there exists x such that x is Socrates and x is a man" and *there exists* is a quantifier while *x* is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.

**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 makes a close link between model theory that deals with what is true in different models, and proof theory that studies what can be formally proven in particular formal systems.

In mathematics, **model theory** is the study of classes of mathematical structures from the perspective of mathematical logic. The objects of study are models of theories in a formal language. A set of sentences in a formal language is one of the components that form a **theory**. A **model** of a theory is a structure that satisfies the sentences of that theory.

In logic and mathematics **second-order logic** is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory.

In mathematical logic, the **Löwenheim–Skolem theorem** is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem.

In mathematics, in set theory, the **constructible universe**, denoted L, is a particular class of sets that can be described entirely in terms of simpler sets. `L` is the union of the **constructible hierarchy**`L`_{α} . It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". In this, he proved that the constructible universe is an inner model of ZF set theory, and also that the axiom of choice and the generalized continuum hypothesis are true in the constructible universe. This shows that both propositions are consistent with the basic axioms of set theory, if ZF itself is consistent. Since many other theorems only hold in systems in which one or both of the propositions is true, their consistency is an important result.

**Metalogic** is the study of the metatheory of logic. Whereas *logic* studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived *about* the languages and systems that are used to express truths.

**Thoralf Albert Skolem** was a Norwegian mathematician who worked in mathematical logic and set theory.

In model theory, a branch of mathematical logic, two structures *M* and *N* of the same signature *σ* are called **elementarily equivalent** if they satisfy the same first-order *σ*-sentences.

In mathematical logic, a theory is **categorical** if it has exactly one model. Such a theory can be viewed as *defining* its model, uniquely characterizing its structure.

In mathematical logic and philosophy, **Skolem's paradox** is a seeming contradiction that arises from the downward Löwenheim–Skolem theorem. Thoralf Skolem (1922) was the first to discuss the seemingly contradictory aspects of the theorem, and to discover the relativity of set-theoretic notions now known as non-absoluteness. Although it is not an actual antinomy like Russell's paradox, the result is typically called a paradox, and was described as a "paradoxical state of affairs" by Skolem.

In mathematical logic, a **theory** is a set of sentences in a formal language that is closed under logical implication. In most scenarios, a deductive system is first understood from context, after which an element of a theory is then called a theorem of the theory. In many deductive systems there is usually a subset that is called "the set of axioms" of the theory , in which case the deductive system is also called an "axiomatic system". By definition, every axiom is automatically a theorem. A **first-order theory** is a set of first-order sentences (theorems) recursively obtained by the inference rules of the system applied to the set of axioms.

In logic, finite model theory, and computability theory, **Trakhtenbrot's theorem** states that the problem of validity in first-order logic on the class of all finite models is undecidable. In fact, the class of valid sentences over finite models is not recursively enumerable.

In mathematical logic, a **non-standard model of arithmetic** is a model of (first-order) Peano arithmetic that contains non-standard numbers. The term **standard model of arithmetic** refers to the standard natural numbers 0, 1, 2, …. The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. A non-standard model is one that has additional elements outside this initial segment. The construction of such models is due to Thoralf Skolem (1934).

An **interpretation** is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called formal semantics.

In mathematical logic, **true arithmetic** is the set of all true statements about the arithmetic of natural numbers. This is the theory associated with the standard model of the Peano axioms in the language of the first-order Peano axioms. True arithmetic is occasionally called Skolem arithmetic, though this term usually refers to the different theory of natural numbers with multiplication.

In model theory, a branch of mathematical logic, the **Łoś–Vaught test** is a criterion for a theory to be complete, unable to be augmented without becoming inconsistent. For theories in classical logic, this means that for every sentence the theory contains either the sentence or its negation but not both.

In mathematical logic the **Löwenheim number** of an abstract logic is the smallest cardinal number for which a weak downward Löwenheim–Skolem theorem holds. They are named after Leopold Löwenheim, who proved that these exist for a very broad class of logics.

This is a **glossary of set theory**.

- Boolos, George; Jeffrey, Richard; Burgess, John (2004).
*Computability and Logic**(fourth ed.). Cambridge University Press.* - Chang, C.C.; Keisler, H. Jerome (1989).
*Model Theory*(third ed.). Elsevier. ISBN 0-7204-0692-7. - Dawson, John W. junior (1993). "The compactness of first-order logic: From Gödel to Lindström".
*History and Philosophy of Logic*.**14**: 15–37. doi:10.1080/01445349308837208. - Hodges, Wilfrid (1993).
*Model theory*. Cambridge University Press. ISBN 0-521-30442-3. - Marker, David (2002).
*Model Theory: An Introduction*. Graduate Texts in Mathematics 217. Springer. ISBN 0-387-98760-6. - Truss, John K. (1997).
*Foundations of Mathematical Analysis*. Oxford University Press. ISBN 0-19-853375-6.

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