Veblen hierarchy
E255569
construction in proof theory
hierarchy of ordinal functions
mathematical concept
ordinal notation system
The Veblen hierarchy is a transfinite sequence of ordinal functions used in mathematical logic and set theory to systematically generate and classify very large countable ordinals.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Veblen hierarchy canonical | 4 |
| Veblen function φ_α | 1 |
| Veblen functions | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2314440 — 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: Veblen hierarchy Context triple: [Oswald Veblen, notableWork, Veblen hierarchy]
-
A.
Hicks–Kaldor compensation criterion
The Hicks–Kaldor compensation criterion is an economic efficiency test stating that a policy change is desirable if those who gain could in principle compensate those who lose and still be better off, regardless of whether compensation actually occurs.
-
B.
Kaldorian cumulative causation
Kaldorian cumulative causation is an economic theory proposing that growth and industrial development are driven by self-reinforcing feedback loops, where initial advantages in productivity, demand, or exports lead to further gains and regional divergence.
-
C.
Hicksian demand
Hicksian demand is a concept in microeconomics that describes how a consumer’s demand for goods changes when prices vary while holding utility (satisfaction) constant, often used in welfare and consumer theory.
-
D.
Pareto efficiency
Pareto efficiency is an economic concept describing an allocation of resources where no individual can be made better off without making someone else worse off.
-
E.
The Theory and Measurement of Demand
The Theory and Measurement of Demand is a foundational economics book by Henry Schultz that rigorously develops statistical and mathematical methods for estimating consumer demand.
- 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: Veblen hierarchy Target entity description: The Veblen hierarchy is a transfinite sequence of ordinal functions used in mathematical logic and set theory to systematically generate and classify very large countable ordinals.
-
A.
Hicks–Kaldor compensation criterion
The Hicks–Kaldor compensation criterion is an economic efficiency test stating that a policy change is desirable if those who gain could in principle compensate those who lose and still be better off, regardless of whether compensation actually occurs.
-
B.
Kaldorian cumulative causation
Kaldorian cumulative causation is an economic theory proposing that growth and industrial development are driven by self-reinforcing feedback loops, where initial advantages in productivity, demand, or exports lead to further gains and regional divergence.
-
C.
Hicksian demand
Hicksian demand is a concept in microeconomics that describes how a consumer’s demand for goods changes when prices vary while holding utility (satisfaction) constant, often used in welfare and consumer theory.
-
D.
Pareto efficiency
Pareto efficiency is an economic concept describing an allocation of resources where no individual can be made better off without making someone else worse off.
-
E.
The Theory and Measurement of Demand
The Theory and Measurement of Demand is a foundational economics book by Henry Schultz that rigorously develops statistical and mathematical methods for estimating consumer demand.
- F. None of above. chosen
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
construction in proof theory
ⓘ
hierarchy of ordinal functions ⓘ mathematical concept ⓘ ordinal notation system ⓘ |
| appearsIn |
ordinal analysis of formal theories
ⓘ
research on large countable ordinals ⓘ |
| basedOn |
normal functions on ordinals
ⓘ
ordinal arithmetic ⓘ |
| canReach |
Feferman–Schütte ordinal Γ₀
ⓘ
linked to:
Feferman–Schütte ordinal
|
| cardinality | countable range at each finite stage ⓘ |
| codomain | ordinal numbers ⓘ |
| constructionDetail |
φ_{α+1} enumerates common fixed points of φ_α
ⓘ
φ_λ for limit λ is defined via simultaneous recursion over all φ_α with α<λ ⓘ |
| constructionMethod | iterating fixed-point operations on normal ordinal functions ⓘ |
| defines |
Veblen functions
ⓘ
linked to:
Veblen hierarchy
|
| domain | ordinal numbers ⓘ |
| field |
mathematical logic
ⓘ
proof theory ⓘ set theory ⓘ |
| generalizes |
Cantor normal form
ⓘ
iterated exponential functions on ordinals ⓘ |
| hasPart |
Veblen function φ_α
ⓘ
linked to:
Veblen hierarchy
|
| infiniteSequence | indexed by ordinal parameters ⓘ |
| introducedIn | early 20th century ⓘ |
| lowestLevel | φ_0(β) = ω^β ⓘ |
| namedAfter | Oswald Veblen ⓘ |
| notation | φ_α(β) ⓘ |
| property |
each Veblen function is a normal function
ⓘ
functions are strictly increasing and continuous in the ordinal topology ⓘ |
| purpose |
to classify large countable ordinals
ⓘ
to generate large countable ordinals ⓘ |
| relatedTo |
Bachmann–Howard ordinal
ⓘ
Feferman–Schütte ordinal Γ₀ ⓘ
linked to:
Feferman–Schütte ordinal
epsilon numbers ⓘ ordinal collapsing functions ⓘ |
| topic |
fixed points of normal functions
ⓘ
large countable ordinals ⓘ transfinite induction ⓘ |
| usedFor |
analyzing proof-theoretic ordinals
ⓘ
describing fixed points of ordinal functions ⓘ ordinal notation beyond ε₀ ⓘ |
| usedIn | classification of proof-theoretic strength of theories ⓘ |
| uses | transfinite recursion ⓘ |
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: Veblen hierarchy Description of subject: The Veblen hierarchy is a transfinite sequence of ordinal functions used in mathematical logic and set theory to systematically generate and classify very large countable ordinals.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Veblen hierarchy
linked to: Veblen hierarchy