Bachmann–Howard ordinal
E890897
The Bachmann–Howard ordinal is a large countable ordinal that serves as a key benchmark in proof theory, marking the strength of powerful formal systems extending predicative arithmetic.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Bachmann–Howard ordinal canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T10829113 — 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: Bachmann–Howard ordinal Context triple: [Veblen hierarchy, relatedTo, Bachmann–Howard ordinal]
-
A.
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.
-
B.
Cantor normal form
Cantor normal form is a canonical way of expressing any ordinal number as a finite sum of decreasing powers of the first infinite ordinal ω with natural number coefficients.
-
C.
Graham's number
Graham's number is an extraordinarily large number that arose in a problem in Ramsey theory and became famous as one of the largest numbers ever used in a serious mathematical proof.
-
D.
Ackermann function
The Ackermann function is a classic example of a computable function that grows faster than any primitive recursive function, often used in theoretical computer science to illustrate extreme computational complexity.
-
E.
Conway chained arrow notation
Conway chained arrow notation is a mathematical system of hyper-operator-style notation introduced by John Horton Conway to concisely represent extremely large numbers.
- 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: Bachmann–Howard ordinal Target entity description: The Bachmann–Howard ordinal is a large countable ordinal that serves as a key benchmark in proof theory, marking the strength of powerful formal systems extending predicative arithmetic.
-
A.
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.
-
B.
Cantor normal form
Cantor normal form is a canonical way of expressing any ordinal number as a finite sum of decreasing powers of the first infinite ordinal ω with natural number coefficients.
-
C.
Graham's number
Graham's number is an extraordinarily large number that arose in a problem in Ramsey theory and became famous as one of the largest numbers ever used in a serious mathematical proof.
-
D.
Ackermann function
The Ackermann function is a classic example of a computable function that grows faster than any primitive recursive function, often used in theoretical computer science to illustrate extreme computational complexity.
-
E.
Conway chained arrow notation
Conway chained arrow notation is a mathematical system of hyper-operator-style notation introduced by John Horton Conway to concisely represent extremely large numbers.
- F. None of above. chosen
Statements (30)
| Predicate | Object |
|---|---|
| instanceOf | countable ordinal ⓘ |
| appearsIn | ordinal analyses of theories stronger than predicative analysis ⓘ |
| characterizes | the proof-theoretic strength of powerful systems extending predicative arithmetic ⓘ |
| comparedWith | proof-theoretic ordinals of systems of analysis ⓘ |
| field |
mathematical logic
ⓘ
proof theory ⓘ set theory ⓘ |
| greaterThan |
Feferman–Schütte ordinal Γ₀
ⓘ
linked to:
Feferman–Schütte ordinal
|
| hasNotation | often denoted by ψ(Ω_ω) or similar collapsing-function terms ⓘ |
| hasProperty |
can be defined without reference to uncountable sets
ⓘ
can be described using collapsing functions ⓘ countable but not representable by simple Veblen hierarchy alone ⓘ is a limit of smaller recursive ordinals ⓘ serves as a boundary for many systems of second-order arithmetic ⓘ strictly less than any uncountable ordinal ⓘ well-founded under ordinal comparison ⓘ |
| is |
a benchmark for the strength of formal systems
ⓘ
a key example in the study of large countable ordinals ⓘ a recursive ordinal ⓘ a well-ordered type of a recursive well-ordering ⓘ |
| namedAfter |
Heinz Bachmann
ⓘ
William Alvin Howard ⓘ |
| relatedTo |
Veblen hierarchy
ⓘ
impredicative comprehension ⓘ ordinal collapsing functions ⓘ predicative arithmetic ⓘ |
| usedIn |
analysis of impredicative theories
ⓘ
ordinal analysis of formal theories ⓘ |
| usedToMeasure |
consistency strength of certain formal systems
ⓘ
proof-theoretic strength of subsystems of second-order arithmetic ⓘ |
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: Bachmann–Howard ordinal Description of subject: The Bachmann–Howard ordinal is a large countable ordinal that serves as a key benchmark in proof theory, marking the strength of powerful formal systems extending predicative arithmetic.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.