Rosser trick
E943471
The Rosser trick is a refinement of Gödel’s incompleteness proof that avoids using ω-consistency by constructing a self-referential sentence asserting that a shorter proof of its negation exists.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Rosser trick canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T11736264 — 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 trick Context triple: [Barkley Rosser, notableWork, Rosser trick]
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A.
Tarski–Mostowski–Robinson theorem
The Tarski–Mostowski–Robinson theorem is a fundamental result in model theory that characterizes when a class of structures is first-order axiomatizable, linking definability properties with closure under ultraproducts and isomorphisms.
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B.
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.
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C.
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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D.
Kleene’s recursion theorem
Kleene’s recursion theorem is a fundamental result in computability theory that guarantees the existence of self-referential programs, allowing a program to effectively obtain and use its own description.
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E.
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.
- 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 trick Target entity description: The Rosser trick is a refinement of Gödel’s incompleteness proof that avoids using ω-consistency by constructing a self-referential sentence asserting that a shorter proof of its negation exists.
-
A.
Tarski–Mostowski–Robinson theorem
The Tarski–Mostowski–Robinson theorem is a fundamental result in model theory that characterizes when a class of structures is first-order axiomatizable, linking definability properties with closure under ultraproducts and isomorphisms.
-
B.
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.
-
C.
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.
-
D.
Kleene’s recursion theorem
Kleene’s recursion theorem is a fundamental result in computability theory that guarantees the existence of self-referential programs, allowing a program to effectively obtain and use its own description.
-
E.
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.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
method in mathematical logic
ⓘ
proof technique ⓘ refinement of Gödel’s incompleteness proof ⓘ |
| appearsIn |
expositions of incompleteness theorems
ⓘ
standard textbooks on mathematical logic ⓘ |
| appliesTo |
formal arithmetic
ⓘ
recursively axiomatized theories ⓘ sufficiently strong consistent theories ⓘ |
| assumes | consistency ⓘ |
| avoidsAssumption | ω-consistency ⓘ |
| basedOn |
Gödel numbering
ⓘ
diagonalization ⓘ |
| constructs |
Rosser sentence
ⓘ
self-referential sentence ⓘ |
| differsFrom | Gödel’s original construction by using proof-length comparison ⓘ |
| ensures |
if theory is consistent, Rosser sentence is undecidable
ⓘ
no proof of the Rosser sentence exists without a shorter proof of its negation ⓘ no proof of the negation exists without a shorter proof of the sentence ⓘ |
| field |
mathematical logic
ⓘ
metamathematics ⓘ proof theory ⓘ |
| generalizes |
Gödel’s first incompleteness theorem
ⓘ
linked to:
Gödel's incompleteness theorems
|
| hasConsequence | even mere consistency implies incompleteness for strong enough theories ⓘ |
| historicalContext | developed after Gödel’s 1931 incompleteness theorems ⓘ |
| influenced |
later work on provability logic
ⓘ
refinements of incompleteness theorems ⓘ |
| involves |
coding of finite sequences
ⓘ
formalization of provability within arithmetic ⓘ partial truth definitions for arithmetic ⓘ primitive recursive relations ⓘ |
| keyIdea |
compare lengths of proofs of a sentence and its negation
ⓘ
encode proof minimality into the sentence ⓘ sentence asserts existence of a shorter proof of its negation ⓘ |
| logicalProperty | yields a sentence independent of the theory if the theory is consistent ⓘ |
| namedAfter | J. Barkley Rosser ⓘ |
| namedInHonorOf | J. Barkley Rosser ⓘ |
| relatesTo | Gödel sentence ⓘ |
| requires |
arithmetization of syntax
ⓘ
effective proof predicate ⓘ |
| strengthens | hypotheses of Gödel’s original incompleteness proof ⓘ |
| typicalDomain |
first-order arithmetic
ⓘ
theories extending Robinson arithmetic ⓘ |
| usedFor |
proving incompleteness of Peano arithmetic
ⓘ
proving incompleteness of stronger arithmetic theories ⓘ removing ω-consistency from incompleteness assumptions ⓘ |
| usedInProofOf |
Rosser’s incompleteness theorem
ⓘ
linked to:
Rosser’s trick in incompleteness proofs
first incompleteness theorem without ω-consistency ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Rosser trick Description of subject: The Rosser trick is a refinement of Gödel’s incompleteness proof that avoids using ω-consistency by constructing a self-referential sentence asserting that a shorter proof of its negation exists.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.