Kleene hierarchy
E601581
classification of predicates
classification of sets
concept in arithmetic hierarchy theory
concept in mathematical logic
concept in recursion theory
hierarchy of definability
The Kleene hierarchy is a classification of sets and predicates in arithmetic and recursion theory based on their definability and complexity, introduced by logician Stephen Kleene.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Kleene hierarchy canonical | 3 |
| arithmetical hierarchy | 3 |
How this entity was disambiguated
This entity first appeared as the object of triple T6594127 — 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 hierarchy Context triple: [Stephen Kleene, knownFor, Kleene hierarchy]
-
A.
Kleene numbering
Kleene numbering is a method in computability theory for effectively assigning natural numbers to partial recursive functions, refining Gödel numbering to study algorithmic properties of functions.
-
B.
Chomsky hierarchy
The Chomsky hierarchy is a classification of formal grammars into four types that correspond to increasing levels of generative power and computational complexity in formal language theory.
-
C.
Kripke–Platek set theory
Kripke–Platek set theory is a weaker, predicative subsystem of Zermelo–Fraenkel set theory focused on sets that are explicitly constructible and often used in the study of admissible sets and recursion theory.
-
D.
Feferman–Schütte ordinal
The Feferman–Schütte ordinal is a large countable ordinal that marks the proof-theoretic strength of predicative arithmetic and analysis, serving as a key boundary in ordinal analysis and foundations of mathematics.
-
E.
Rice's theorem
Rice's theorem is a fundamental result in computability theory stating that any non-trivial semantic property of the language recognized by a Turing machine is undecidable.
- 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 hierarchy Target entity description: The Kleene hierarchy is a classification of sets and predicates in arithmetic and recursion theory based on their definability and complexity, introduced by logician Stephen Kleene.
-
A.
Kleene numbering
Kleene numbering is a method in computability theory for effectively assigning natural numbers to partial recursive functions, refining Gödel numbering to study algorithmic properties of functions.
-
B.
Chomsky hierarchy
The Chomsky hierarchy is a classification of formal grammars into four types that correspond to increasing levels of generative power and computational complexity in formal language theory.
-
C.
Kripke–Platek set theory
Kripke–Platek set theory is a weaker, predicative subsystem of Zermelo–Fraenkel set theory focused on sets that are explicitly constructible and often used in the study of admissible sets and recursion theory.
-
D.
Feferman–Schütte ordinal
The Feferman–Schütte ordinal is a large countable ordinal that marks the proof-theoretic strength of predicative arithmetic and analysis, serving as a key boundary in ordinal analysis and foundations of mathematics.
-
E.
Rice's theorem
Rice's theorem is a fundamental result in computability theory stating that any non-trivial semantic property of the language recognized by a Turing machine is undecidable.
- F. None of above. chosen
Statements (39)
| Predicate | Object |
|---|---|
| instanceOf |
classification of predicates
ⓘ
classification of sets ⓘ concept in arithmetic hierarchy theory ⓘ concept in mathematical logic ⓘ concept in recursion theory ⓘ hierarchy of definability ⓘ |
| appliesTo |
arithmetical predicates
ⓘ
subsets of natural numbers ⓘ |
| areaOfApplication |
foundations of mathematics
ⓘ
proof theory ⓘ theory of computation ⓘ |
| basedOn |
arithmetical definability
ⓘ
complexity of formulas ⓘ quantifier complexity ⓘ |
| characterizes |
complexity of definable predicates over arithmetic
ⓘ
complexity of definable sets of integers ⓘ |
| concerns |
effective definability
ⓘ
logical complexity of definitions ⓘ |
| describes |
classification of predicates by definability
ⓘ
classification of sets by definability ⓘ |
| field |
arithmetical hierarchy
ⓘ
linked to:
Kleene hierarchy
mathematical logic ⓘ recursion theory ⓘ |
| hasLevel | finite levels indexed by natural numbers ⓘ |
| influenced |
classification schemes in descriptive set theory
ⓘ
later work on hierarchies in recursion theory ⓘ |
| introducedBy |
Stephen Cole Kleene
ⓘ
linked to:
Stephen Kleene
|
| namedAfter |
Stephen Cole Kleene
ⓘ
linked to:
Stephen Kleene
|
| organizes |
predicates by increasing definitional complexity
ⓘ
sets by increasing definitional complexity ⓘ |
| relatedTo |
Kleene normal form theorem
ⓘ
linked to:
Kleene’s normal form theorem
Turing degrees ⓘ analytical hierarchy ⓘ arithmetical hierarchy ⓘ
linked to:
Kleene hierarchy
effective descriptive set theory ⓘ |
| usesConcept |
arithmetical formula
ⓘ
partial recursive function ⓘ quantifier alternation ⓘ recursive function ⓘ |
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 hierarchy Description of subject: The Kleene hierarchy is a classification of sets and predicates in arithmetic and recursion theory based on their definability and complexity, introduced by logician Stephen Kleene.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Kleene
linked to: Kleene hierarchy
linked to: Kleene hierarchy
linked to: Kleene hierarchy