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First-order logic

predicate logicfirst-orderpredicate calculus
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.
First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.

Model theory

modelmodelsmodel-theoretic
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.
Model theory in higher-order logics or infinitary logics is hampered by the fact that completeness and compactness do not in general hold for these logics.

Mathematical logic

formal logicsymbolic logiclogic
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.
Gödel used the completeness theorem to prove the compactness theorem, demonstrating the finitary nature of first-order logical consequence.

Lindström's theorem

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.
In mathematical logic, Lindström's theorem (named after Swedish logician Per Lindström, who published it in 1969) states that first-order logic is the strongest logic (satisfying certain conditions, e.g. closure under classical negation) having both the (countable) compactness property and the (downward) Löwenheim–Skolem property.

Löwenheim–Skolem theorem

(downward) Löwenheim–Skolem propertydownward Löwenheim–Skolem theoremLöwenheim-Skolem theorem
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. 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).
The (downward) Löwenheim–Skolem theorem is one of the two key properties, along with the compactness theorem, that are used in Lindström's theorem to characterize first-order logic.

Ultraproduct

ultrapowerŁoś' theoremultrapowers
One of those proofs relies on ultraproducts hinging on the axiom of choice as follows:
Some striking applications of ultraproducts include very elegant proofs of the compactness theorem and the completeness theorem, Keisler's ultrapower theorem, which gives an algebraic characterization of the semantic notion of elementary equivalence, and the Robinson-Zakon presentation of the use of superstructures and their monomorphisms to construct nonstandard models of analysis, leading to the growth of the area of non-standard analysis, which was pioneered (as an application of the compactness theorem) by Abraham Robinson.

Peano axioms

Peano arithmeticfirst-order arithmeticarithmetic
So, for instance, there are nonstandard models of Peano arithmetic with uncountably many 'natural numbers'.
Although the usual natural numbers satisfy the axioms of PA, there are other models as well (called "non-standard models"); the compactness theorem implies that the existence of nonstandard elements cannot be excluded in first-order logic.

Gödel's completeness theorem

completeness theoremcompletenesscomplete
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.
The completeness theorem and the compactness theorem are two cornerstones of first-order logic.

List of Boolean algebra topics

Boolean algebra topicslist
List of Boolean algebra topics
Compactness theorem

Barwise compactness theorem

Barwise compactness theorem
In mathematical logic, the Barwise compactness theorem, named after Jon Barwise, is a generalization of the usual compactness theorem for first-order logic to a certain class of infinitary languages.

Subset

supersetproper subsetinclusion
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.

Consistency

consistentinconsistentlogically consistent
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.

Propositional calculus

propositional logicpropositionalsentential logic
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; hence, the theorem's name.

Tychonoff's theorem

central theoremcompactness theoremtheorem
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; hence, the theorem's name.

Compact space

compactcompact setcompactness
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; hence, the theorem's name.

Stone space

Stone space'''.Stone's theorem
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; hence, the theorem's name.

Finite intersection property

strong finite intersection propertycompactness
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.

Kurt Gödel

GödelGödel, Kurt Gödel, Kurt
Kurt Gödel proved the countable compactness theorem in 1930.

Anatoly Maltsev

A.I. Mal'tsevMalcevMaltsev Prize
Anatoly Maltsev proved the uncountable case in 1936.

Field (mathematics)

fieldfieldsfield theory
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.

Characteristic (algebra)

characteristicprime fieldcharacteristic zero
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.

Cardinality

cardinalitiesnumber of elementssize
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).

Cardinal number

cardinalcardinal numberscardinality
To achieve this, let T be the initial theory and let κ be any cardinal number.

Non-standard analysis

nonstandard analysisnon-standardinfinitesimals
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.