Rosser’s trick in incompleteness proofs
E943475
Rosser’s trick in incompleteness proofs is a refinement of Gödel’s incompleteness argument that strengthens the result by avoiding the need for the assumption that the underlying formal system is ω-consistent.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Rosser’s trick in incompleteness proofs canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11736279 — 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: Rosser’s trick in incompleteness proofs Context triple: [Barkley Rosser, notableConcept, Rosser’s trick in incompleteness proofs]
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A.
Gentzen’s consistency proof for arithmetic
Gentzen’s consistency proof for arithmetic is a landmark 1930s result in proof theory that established the consistency of Peano arithmetic using transfinite induction up to the ordinal ε₀.
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B.
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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C.
arithmetization of syntax
Arithmetization of syntax is a method in mathematical logic that encodes formal language expressions and proofs as natural numbers so that syntactic properties can be studied using arithmetic.
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D.
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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E.
Löb's theorem
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.
- 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: Rosser’s trick in incompleteness proofs Target entity description: Rosser’s trick in incompleteness proofs is a refinement of Gödel’s incompleteness argument that strengthens the result by avoiding the need for the assumption that the underlying formal system is ω-consistent.
-
A.
Gentzen’s consistency proof for arithmetic
Gentzen’s consistency proof for arithmetic is a landmark 1930s result in proof theory that established the consistency of Peano arithmetic using transfinite induction up to the ordinal ε₀.
-
B.
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.
-
C.
arithmetization of syntax
Arithmetization of syntax is a method in mathematical logic that encodes formal language expressions and proofs as natural numbers so that syntactic properties can be studied using arithmetic.
-
D.
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.
-
E.
Löb's theorem
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.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
method in mathematical logic
ⓘ
proof technique ⓘ refinement of Gödel’s incompleteness argument ⓘ |
| appearsIn |
expositions of incompleteness theorems
ⓘ
standard textbooks on mathematical logic ⓘ |
| appliesTo |
Peano arithmetic
NERFINISHED
ⓘ
formal arithmetic ⓘ sufficiently strong recursively axiomatized theories ⓘ |
| assumptionRequired | simple consistency ⓘ |
| assumptionWeakened | ω-consistency ⓘ |
| avoids | use of ω-consistency in the incompleteness proof ⓘ |
| category |
self-reference constructions in logic
ⓘ
techniques in incompleteness proofs ⓘ |
| consequence | every consistent, effectively axiomatized, sufficiently strong theory is incomplete ⓘ |
| contrastWith | Gödel’s original use of ω-consistency ⓘ |
| ensures | existence of a sentence undecidable under mere consistency ⓘ |
| field |
mathematical logic
ⓘ
metamathematics ⓘ proof theory ⓘ |
| formalProperty | constructs a sentence undecidable if the theory is consistent ⓘ |
| historicalContext | developed after Gödel’s 1931 incompleteness theorems ⓘ |
| influenced | later refinements of incompleteness theorems ⓘ |
| involves |
construction of a Rosser sentence
ⓘ
formalization of proof length comparisons ⓘ |
| keyIdea |
encode a statement that any proof of the sentence has a shorter proof of its negation
ⓘ
use of a sentence comparing proofs of itself and its negation ⓘ |
| logicalFramework | first-order arithmetic ⓘ |
| modifies | Gödel’s original self-referential sentence construction ⓘ |
| namedAfter | J. Barkley Rosser NERFINISHED ⓘ |
| relatedConcept |
Gödel sentence
NERFINISHED
ⓘ
Rosser sentence ⓘ |
| relatedTo |
Hilbert’s program
NERFINISHED
ⓘ
limitations of formal systems ⓘ |
| reliesOn |
Gödel numbering
NERFINISHED
ⓘ
arithmetization of syntax ⓘ representability of provability in arithmetic ⓘ |
| requires |
ability to formalize provability predicates
ⓘ
ability to reason about proofs within the theory ⓘ |
| strengthens | Gödel’s first incompleteness theorem NERFINISHED ⓘ |
| typicalAssumptionOnTheory |
recursively enumerable axioms
GENERATED
ⓘ
sufficient arithmetic strength to represent primitive recursive functions GENERATED ⓘ |
| usedFor |
avoiding the ω-consistency assumption
ⓘ
proving incompleteness results ⓘ |
| usedIn | Rosser’s incompleteness theorem NERFINISHED ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Rosser’s trick in incompleteness proofs Description of subject: Rosser’s trick in incompleteness proofs is a refinement of Gödel’s incompleteness argument that strengthens the result by avoiding the need for the assumption that the underlying formal system is ω-consistent.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.