Löb's theorem
E353392
Löb's theorem is a fundamental result in mathematical logic that characterizes when a sufficiently strong formal system can prove statements about its own provability, closely refining the insights of Gödel’s incompleteness theorems.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Löb's theorem canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T3390153 — 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: Löb's theorem Context triple: [Gödel's incompleteness theorems, relatedTo, Löb's theorem]
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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.
Gödel's incompleteness theorems
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.
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C.
Kripke fixed-point theory of truth
The Kripke fixed-point theory of truth is a semantic framework developed by Saul Kripke that uses partial truth predicates and fixed points to consistently handle self-referential sentences and semantic paradoxes like the liar paradox.
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D.
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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E.
Gödel's ontological proof
Gödel's ontological proof is a formal, modal-logic-based argument for the existence of God that rigorously develops and refines earlier ontological arguments within a precise axiomatic framework.
- 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: Löb's theorem Target entity description: Löb's theorem is a fundamental result in mathematical logic that characterizes when a sufficiently strong formal system can prove statements about its own provability, closely refining the insights of Gödel’s incompleteness theorems.
-
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.
Gödel's incompleteness theorems
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.
-
C.
Kripke fixed-point theory of truth
The Kripke fixed-point theory of truth is a semantic framework developed by Saul Kripke that uses partial truth predicates and fixed points to consistently handle self-referential sentences and semantic paradoxes like the liar paradox.
-
D.
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.
-
E.
Gödel's ontological proof
Gödel's ontological proof is a formal, modal-logic-based argument for the existence of God that rigorously develops and refines earlier ontological arguments within a precise axiomatic framework.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in mathematical logic ⓘ |
| appliesTo | sufficiently strong, effectively axiomatized, consistent theories extending Peano arithmetic ⓘ |
| assumes | Hilbert–Bernays derivability conditions for the provability predicate ⓘ |
| assumption |
the provability predicate correctly represents theoremhood in the theory
ⓘ
the theory is capable of representing a sufficient amount of arithmetic ⓘ |
| category |
theorems about formal theories
ⓘ
theorems about provability predicates ⓘ |
| concerns |
formal theories of arithmetic
ⓘ
provability in formal systems ⓘ self-referential statements ⓘ |
| doesNotRequire | ω-consistency ⓘ |
| field |
mathematical logic
ⓘ
metamathematics ⓘ proof theory ⓘ |
| formalStatementPattern | for all sentences φ, if T ⊢ Prov_T(⌜φ⌝) → φ then T ⊢ φ ⓘ |
| hasAlternativeName | Löb's rule in arithmetic context ⓘ |
| hasConsequence |
constraints on self-referential provability assertions
ⓘ
fixed points for provability operators in arithmetic ⓘ limitations on internal reflection principles in arithmetic ⓘ |
| holdsIn |
Peano arithmetic
ⓘ
many recursively axiomatizable extensions of Peano arithmetic ⓘ |
| implies |
if a theory proves that its own provability of φ implies φ, then it proves φ
ⓘ
no nontrivial theory satisfying the derivability conditions can prove its own consistency ⓘ |
| influenced | research on reflection and self-reference in logic ⓘ |
| inspired | Gödel–Löb provability logic (GL) ⓘ |
| isEquivalentTo | modal logic axiom GL: □(□p → p) → □p under arithmetical interpretation ⓘ |
| namedAfter | Martin Hugo Löb ⓘ |
| refines | Gödel's incompleteness theorems ⓘ |
| relatedTo |
Gödel's first incompleteness theorem
ⓘ
linked to:
Gödel's incompleteness theorems
Gödel's second incompleteness theorem ⓘ
linked to:
Gödel's incompleteness theorems
Hilbert–Bernays derivability conditions ⓘ formalized provability logic ⓘ reflection principles in arithmetic ⓘ self-reference in formal systems ⓘ |
| requires |
Gödel numbering of formulas and proofs
ⓘ
formalization of syntactic notions inside arithmetic ⓘ |
| standardFormulation | If T ⊢ Prov_T(⌜φ⌝) → φ, then T ⊢ φ ⓘ |
| strengthens | Gödel's second incompleteness theorem in terms of provability conditions ⓘ |
| topicOf |
metalogical investigations of arithmetic
ⓘ
provability logic ⓘ |
| usedIn |
analysis of formal verification of proofs
ⓘ
foundations of modal provability logic ⓘ study of self-referential paradoxes in arithmetic ⓘ |
| uses |
arithmetized provability predicate
ⓘ
fixed-point (diagonal) lemma ⓘ |
| yearProved | 1955 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Löb's theorem Description of subject: Löb's theorem is a fundamental result in mathematical logic that characterizes when a sufficiently strong formal system can prove statements about its own provability, closely refining the insights of Gödel’s incompleteness theorems.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.