Gödel's incompleteness theorems
E71396
Gödel's incompleteness theorems are two fundamental results in mathematical logic showing that any sufficiently powerful, consistent formal system cannot prove all true statements about arithmetic, and cannot prove its own consistency.
All labels observed (9)
How this entity was disambiguated
This entity first appeared as the object of triple T568429 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Gödel's incompleteness theorems Context triple: [liar paradox, relatedTo, Gödel's incompleteness theorems]
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A.
Tarski's undefinability theorem
Tarski's undefinability theorem is a fundamental result in mathematical logic showing that, in sufficiently strong formal systems, the notion of truth for the language of the system cannot be defined within that same language.
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B.
Hilbert’s program
Hilbert’s program was an influential early-20th-century initiative in the foundations of mathematics that sought to formalize all of mathematics and prove its consistency using finitistic methods.
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C.
Remarks on the Foundations of Mathematics
Remarks on the Foundations of Mathematics is a posthumously published collection of Ludwig Wittgenstein’s later writings that critically examines the nature of mathematical truth, proof, and practice from a philosophical and language-centered perspective.
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D.
Church–Turing thesis
The Church–Turing thesis is a foundational principle in computability theory stating that any function that can be effectively computed by an algorithm can be computed by a Turing machine (or equivalently by other formal models of computation).
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E.
On Computable Numbers with an Application to the Entscheidungsproblem
"On Computable Numbers, with an Application to the Entscheidungsproblem" is Alan Turing’s landmark 1936 paper that introduced the Turing machine model and founded the formal study of computability and the limits of algorithmic decision procedures.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Gödel's incompleteness theorems Target entity description: Gödel's incompleteness theorems are two fundamental results in mathematical logic showing that any sufficiently powerful, consistent formal system cannot prove all true statements about arithmetic, and cannot prove its own consistency.
-
A.
Tarski's undefinability theorem
Tarski's undefinability theorem is a fundamental result in mathematical logic showing that, in sufficiently strong formal systems, the notion of truth for the language of the system cannot be defined within that same language.
-
B.
Hilbert’s program
Hilbert’s program was an influential early-20th-century initiative in the foundations of mathematics that sought to formalize all of mathematics and prove its consistency using finitistic methods.
-
C.
Remarks on the Foundations of Mathematics
Remarks on the Foundations of Mathematics is a posthumously published collection of Ludwig Wittgenstein’s later writings that critically examines the nature of mathematical truth, proof, and practice from a philosophical and language-centered perspective.
-
D.
Church–Turing thesis
The Church–Turing thesis is a foundational principle in computability theory stating that any function that can be effectively computed by an algorithm can be computed by a Turing machine (or equivalently by other formal models of computation).
-
E.
On Computable Numbers with an Application to the Entscheidungsproblem
"On Computable Numbers, with an Application to the Entscheidungsproblem" is Alan Turing’s landmark 1936 paper that introduced the Turing machine model and founded the formal study of computability and the limits of algorithmic decision procedures.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
metamathematical theorem ⓘ result in mathematical logic ⓘ |
| appliesTo |
Peano arithmetic
ⓘ
Zermelo–Fraenkel set theory ⓘ
surface form:
Zermelo–Fraenkel set theory with Choice
effectively axiomatized theories ⓘ formal axiomatic systems ⓘ recursively axiomatizable theories ⓘ sufficiently strong theories of arithmetic ⓘ |
| assumes | ω-consistency in Gödel's original proof of the first theorem ⓘ |
| author | Kurt Gödel ⓘ |
| concerns |
provability in formal systems
ⓘ
truth in arithmetic ⓘ |
| field |
foundations of mathematics
ⓘ
mathematical logic ⓘ metamathematics ⓘ proof theory ⓘ |
| firstTheoremStates | any consistent, effectively axiomatized theory capable of expressing elementary arithmetic is incomplete ⓘ |
| hasPart |
Gödel's incompleteness theorems
self-linksurface differs
ⓘ
surface form:
Gödel's first incompleteness theorem
Gödel's incompleteness theorems self-linksurface differs ⓘ
surface form:
Gödel's second incompleteness theorem
|
| implies |
a sufficiently strong consistent theory cannot prove its own consistency
ⓘ
existence of true but unprovable statements ⓘ limitations of Hilbert's program ⓘ no complete and consistent extension of Peano arithmetic is recursively axiomatizable ⓘ |
| influenced |
philosophy of logic
ⓘ
philosophy of mathematics ⓘ proof theory ⓘ theory of computation ⓘ |
| laterGeneralizedBy | results using only simple consistency instead of ω-consistency ⓘ |
| namedAfter | Kurt Gödel ⓘ |
| originalLanguage | German ⓘ |
| publishedIn | Monatshefte für Mathematik ⓘ |
| relatedTo |
Church–Turing thesis
ⓘ
Gödel numbering ⓘ Hilbert’s program ⓘ
surface form:
Hilbert's program
Löb's theorem ⓘ Peano arithmetic ⓘ Tarski's undefinability theorem ⓘ |
| requires |
ability to represent basic arithmetic
ⓘ
consistency of the formal system ⓘ effective axiomatization ⓘ |
| secondTheoremStates | no consistent, effectively axiomatized theory capable of expressing elementary arithmetic can prove its own consistency ⓘ |
| showsLimitationOf |
axiomatic method for arithmetic
ⓘ
formalism in mathematics ⓘ |
| status | proven ⓘ |
| usesMethod |
arithmetization of syntax
ⓘ
diagonalization ⓘ self-referential sentences ⓘ |
| yearProved | 1931 ⓘ |
How these facts were elicited
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Input
Subject: Gödel's incompleteness theorems Description of subject: Gödel's incompleteness theorems are two fundamental results in mathematical logic showing that any sufficiently powerful, consistent formal system cannot prove all true statements about arithmetic, and cannot prove its own consistency.
Referenced by (31)
Full triples — surface form annotated when it differs from this entity's canonical label.
On Computable Numbers with an Application to the Entscheidungsproblem
→
relatedTo
→
Gödel's incompleteness theorems
ⓘ
subject surface form:
On Computable Numbers, with an Application to the Entscheidungsproblem
this entity surface form:
Gödel’s incompleteness theorems
this entity surface form:
incompleteness theorems
this entity surface form:
Gödel’s incompleteness theorems
this entity surface form:
Gödel’s Incompleteness Theorems
this entity surface form:
Gödel’s incompleteness theorems
Gödel's incompleteness theorems
→
hasPart
→
Gödel's incompleteness theorems
self-linksurface differs
ⓘ
this entity surface form:
Gödel's first incompleteness theorem
Gödel's incompleteness theorems
→
hasPart
→
Gödel's incompleteness theorems
self-linksurface differs
ⓘ
this entity surface form:
Gödel's second incompleteness theorem
this entity surface form:
Gödel incompleteness theorems
this entity surface form:
Gödel’s incompleteness theorems
this entity surface form:
Gödel incompleteness theorems
this entity surface form:
incompleteness theorems
this entity surface form:
incompleteness theorems
this entity surface form:
Gödel’s incompleteness theorems
this entity surface form:
Gödel’s incompleteness theorems
this entity surface form:
Gödel’s second incompleteness theorem
this entity surface form:
Gödel’s incompleteness theorems
this entity surface form:
Gödel’s incompleteness theorems
this entity surface form:
Kurt Gödel's incompleteness theorems paper
this entity surface form:
Gödel's first incompleteness theorem
this entity surface form:
Gödel's second incompleteness theorem
this entity surface form:
Gödel incompleteness theorems