Kleene–Rosser paradox
E943473
The Kleene–Rosser paradox is a logical contradiction in untyped lambda calculus that demonstrated the inconsistency of Alonzo Church’s original formulation of the system.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Kleene–Rosser paradox canonical | 2 |
| Kleene–Rosser paradox in lambda calculus | 1 |
| Kleene–Rosser system | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11736266 — 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: Kleene–Rosser paradox Context triple: [Barkley Rosser, notableWork, Kleene–Rosser paradox]
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A.
Grelling–Nelson paradox
The Grelling–Nelson paradox is a self-referential logical paradox arising from classifying adjectives as "autological" or "heterological," leading to a contradiction when considering whether "heterological" describes itself.
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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.
Berry paradox
The Berry paradox is a self-referential logical paradox arising from phrases like “the smallest positive integer not definable in under eleven words,” which appears to define exactly such a number while claiming it cannot be defined.
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D.
Russell’s paradox
Russell’s paradox is a foundational logical contradiction in naive set theory that reveals problems with sets that contain themselves, leading to major developments in modern logic and the axiomatization of set theory.
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E.
Barber paradox
The Barber paradox is a self-referential logical puzzle about a barber who shaves all and only those who do not shave themselves, illustrating a contradiction similar to Russell’s paradox.
- 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: Kleene–Rosser paradox Target entity description: The Kleene–Rosser paradox is a logical contradiction in untyped lambda calculus that demonstrated the inconsistency of Alonzo Church’s original formulation of the system.
-
A.
Grelling–Nelson paradox
The Grelling–Nelson paradox is a self-referential logical paradox arising from classifying adjectives as "autological" or "heterological," leading to a contradiction when considering whether "heterological" describes itself.
-
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.
Berry paradox
The Berry paradox is a self-referential logical paradox arising from phrases like “the smallest positive integer not definable in under eleven words,” which appears to define exactly such a number while claiming it cannot be defined.
-
D.
Russell’s paradox
Russell’s paradox is a foundational logical contradiction in naive set theory that reveals problems with sets that contain themselves, leading to major developments in modern logic and the axiomatization of set theory.
-
E.
Barber paradox
The Barber paradox is a self-referential logical puzzle about a barber who shaves all and only those who do not shave themselves, illustrating a contradiction similar to Russell’s paradox.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
logical paradox
ⓘ
result in mathematical logic ⓘ semantic paradox ⓘ |
| appliesTo |
original formulation of Church’s lambda calculus
ⓘ
untyped lambda calculus ⓘ |
| concerns |
beta-reduction properties
ⓘ
equational reasoning in lambda calculus ⓘ impredicative definitions in lambda calculus ⓘ unrestricted lambda abstraction ⓘ |
| contradicts | consistency of Church’s original system ⓘ |
| demonstrates |
existence of a lambda term leading to contradiction
ⓘ
limits of naive formalization of functions ⓘ |
| describes | logical contradiction in untyped lambda calculus ⓘ |
| field |
foundations of mathematics
ⓘ
lambda calculus ⓘ mathematical logic ⓘ proof theory ⓘ |
| formalism | lambda calculus notation ⓘ |
| hasConsequence | original untyped system is inconsistent with certain logical principles ⓘ |
| hasDomain |
formal systems
ⓘ
theory of computation ⓘ |
| historicalPeriod | 1930s ⓘ |
| impact |
influenced later work on consistency of formal systems
ⓘ
led to modification of Church’s original system ⓘ |
| involves | construction of a term that asserts its own non-terminating behavior ⓘ |
| isDiscussedIn |
historical studies of Church’s lambda calculus
ⓘ
literature on lambda calculus foundations ⓘ texts on computability theory ⓘ |
| isExampleOf |
paradox arising from unrestricted abstraction
ⓘ
self-referential construction in logic ⓘ |
| motivated |
development of consistent typed lambda calculi
ⓘ
restrictions on abstraction in lambda calculus ⓘ revisions of Church’s original system ⓘ |
| namedAfter |
J. Barkley Rosser
ⓘ
Stephen Cole Kleene ⓘ
linked to:
Stephen Kleene
|
| relatedTo |
Church’s lambda calculus
ⓘ
linked to:
lambda calculus
Church’s thesis ⓘ
linked to:
Church–Turing thesis
Curry’s paradox ⓘ
linked to:
Curry paradox
Russell’s paradox ⓘ combinatory logic ⓘ inconsistency proofs ⓘ liar paradox ⓘ typed lambda calculus ⓘ |
| shows |
danger of unrestricted self-application in formal systems
ⓘ
inconsistency of Church’s original lambda calculus ⓘ |
| uses |
diagonalization technique
ⓘ
self-referential lambda terms ⓘ |
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: Kleene–Rosser paradox Description of subject: The Kleene–Rosser paradox is a logical contradiction in untyped lambda calculus that demonstrated the inconsistency of Alonzo Church’s original formulation of the system.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Kleene–Rosser paradox
linked to: Kleene–Rosser paradox