Kleene numbering
E446862
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Kleene numbering canonical | 1 |
| Kleene’s recursion theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4492992 — 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 numbering Context triple: [Gödel numbering, relatedConcept, Kleene numbering]
-
A.
Gödel numbering
Gödel numbering is a method in mathematical logic that encodes symbols, formulas, and proofs as unique natural numbers, enabling arithmetic to represent and reason about syntactic statements.
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B.
On Computable Numbers with an Application to the Entscheidungsproblem
"On Computable Numbers, with an Application to the Entscheidungsproblem" is Alan Turing’s landmark 1936 paper that introduced the Turing machine model and founded the formal study of computability and the limits of algorithmic decision procedures.
-
C.
Computability and Unsolvability
Computability and Unsolvability is a classic 1958 textbook by Martin Davis that systematically develops the theory of computable functions and undecidable problems, helping to shape modern computability theory.
-
D.
Peano arithmetic
Peano arithmetic is a formal first-order axiomatic system that captures the basic properties of the natural numbers and underpins much of modern mathematical logic and number theory.
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E.
Church–Turing thesis
The Church–Turing thesis is a foundational principle in computability theory stating that any function that can be effectively computed by an algorithm can be computed by a Turing machine (or equivalently by other formal models of computation).
- 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 numbering Target entity description: 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.
-
A.
Gödel numbering
Gödel numbering is a method in mathematical logic that encodes symbols, formulas, and proofs as unique natural numbers, enabling arithmetic to represent and reason about syntactic statements.
-
B.
On Computable Numbers with an Application to the Entscheidungsproblem
"On Computable Numbers, with an Application to the Entscheidungsproblem" is Alan Turing’s landmark 1936 paper that introduced the Turing machine model and founded the formal study of computability and the limits of algorithmic decision procedures.
-
C.
Computability and Unsolvability
Computability and Unsolvability is a classic 1958 textbook by Martin Davis that systematically develops the theory of computable functions and undecidable problems, helping to shape modern computability theory.
-
D.
Peano arithmetic
Peano arithmetic is a formal first-order axiomatic system that captures the basic properties of the natural numbers and underpins much of modern mathematical logic and number theory.
-
E.
Church–Turing thesis
The Church–Turing thesis is a foundational principle in computability theory stating that any function that can be effectively computed by an algorithm can be computed by a Turing machine (or equivalently by other formal models of computation).
- F. None of above. chosen
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
concept in computability theory
ⓘ
mathematical concept ⓘ numbering of partial recursive functions ⓘ |
| assumption | the set of partial recursive functions is countable ⓘ |
| codomain | partial recursive functions ⓘ |
| context |
arithmetization of syntax and computation
ⓘ
classical recursion theory ⓘ |
| contrastsWith | non-effective or arbitrary enumerations of functions ⓘ |
| domain | natural numbers ⓘ |
| enables |
construction of diagonal functions and fixed-point arguments
ⓘ
definition of computably enumerable sets via indices of partial recursive functions ⓘ definition of many-one and Turing reducibility using indices ⓘ |
| field |
computability theory
ⓘ
mathematical logic ⓘ theoretical computer science ⓘ |
| formalizes | the idea of indexing algorithms by natural numbers ⓘ |
| hasCharacteristic | indices are called Gödel numbers or program numbers ⓘ |
| historicalDevelopment | introduced in the development of recursion theory by Stephen Cole Kleene in the 20th century ⓘ |
| implies | existence of an effective enumeration of partial recursive functions ⓘ |
| isPartOf | the standard framework of classical computability theory ⓘ |
| namedAfter | Stephen Cole Kleene NERFINISHED ⓘ |
| property |
allows uniform enumeration of partial recursive functions
ⓘ
is effective (computable) as a mapping from indices to functions ⓘ is surjective onto the class of partial recursive functions ⓘ |
| purpose |
to effectively assign natural numbers to partial recursive functions
ⓘ
to study algorithmic properties of partial recursive functions ⓘ |
| refines | Gödel numbering ⓘ |
| relatedTo |
Gödel numbering
ⓘ
Kleene normal form theorem NERFINISHED ⓘ acceptable numbering ⓘ effective numbering ⓘ partial recursive function ⓘ recursive function theory ⓘ s-m-n theorem NERFINISHED ⓘ universal partial recursive function ⓘ |
| requires |
a fixed effective coding of finite sequences of natural numbers
ⓘ
a fixed formalism for defining partial recursive functions ⓘ |
| supports | proofs of universality and simulation theorems in recursion theory ⓘ |
| usedFor |
defining acceptable programming systems
ⓘ
defining universal partial recursive functions ⓘ formalizing the notion of algorithm in recursion theory ⓘ proving recursion-theoretic invariance results ⓘ studying reducibility and degrees of unsolvability ⓘ |
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 numbering Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Kleene’s recursion theorem