About: Gödel's completeness theorem

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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: If T is such a theory, and φ is a sentence (in the same language) and any model of T is a model of φ, then there is a (first-order) proof of φ using the statements of T as axioms. One sometimes says this as "anything true is provable".

Attributes	Values
rdf:type	Thing
rdfs:label	Gödel's completeness theorem (en)
rdfs:comment	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: If T is such a theory, and φ is a sentence (in the same language) and any model of T is a model of φ, then there is a (first-order) proof of φ using the statements of T as axioms. One sometimes says this as "anything true is provable". (en)
differentFrom	Gödel's incompleteness theorems
sameAs	Gödel's completeness theorem
dbp:wikiPageUsesTemplate	dbt:Block_indent dbt:Cite_journal dbt:Distinguish dbt:Reflist dbt:Short_description dbt:Mathematical_logic dbt:Metalogic
Subject	Proof theory Works by Kurt Gödel Metatheorems Model theory Theorems in the foundations of mathematics
thumbnail	wiki-commons:Special:FilePath/Completude_logique_premier_ordre.png?width=300
foaf:depiction
gold:hypernym	Theorem
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Gödel's_completeness_theorem?oldid=1073097905&ns=0
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Link from a Wikipage to another Wikipage	Formal system Logical consequence Non-standard model of arithmetic Proof theory Provability logic Well-order Algorithm Arithmetical hierarchy Axiom of choice Boolean prime ideal theorem Semantics Proof theory Works by Kurt Gödel Gisbert Hasenjaeger Juliette Kennedy Metatheorems Model theory Doctor of Philosophy Hilbert system Kurt Gödel Leon Henkin Second-order logic Zermelo–Fraenkel set theory Lindström's theorem Robinson arithmetic Soundness Tennenbaum's theorem Compactness theorem Computable function Computably enumerable Computer Consistency Countable set Herbrand structure Original proof of Gödel's completeness theorem Reverse mathematics First-order logic If and only if Theorems in the foundations of mathematics Group theory Gödel's incompleteness theorems Intuitionistic logic Kripke semantics Modal logic Model theory Monatshefte für Mathematik Peano axioms Tree structure Validity (logic) Stanford Encyclopedia of Philosophy Structure (mathematical logic) Theory (mathematical logic) Isabelle (proof assistant) Löwenheim–Skolem theorem Mathematical logic Natural deduction Thesis
has abstract	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: If T is such a theory, and φ is a sentence (in the same language) and any model of T is a model of φ, then there is a (first-order) proof of φ using the statements of T as axioms. One sometimes says this as "anything true is provable". 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. It was first proved by Kurt Gödel in 1929. It was then simplified when Leon Henkin observed in his Ph.D. thesis that the hard part of the proof can be presented as the Model Existence Theorem (published in 1949). Henkin's proof was simplified by Gisbert Hasenjaeger in 1953. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Gödel's_completeness_theorem
is Wikipage redirect of	Godel's completeness theorem Goedel completeness theorem Gödel completeness theorem Godel completeness theorem Goedel's completeness theorem Goedels completeness theorem Completeness theorem
is Link from a Wikipage to another Wikipage of	Foundations of mathematics List of theorems Non-standard model of arithmetic 1929 in science Automated theorem proving Axiom Axiom of choice Gentzen's consistency proof Gisbert Hasenjaeger

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