Feferman–Schütte ordinal

E513379

countable ordinal large countable ordinal ordinal number proof-theoretic 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.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (2)

Label	Occurrences
Feferman–Schütte ordinal Γ₀	2
Feferman–Schütte ordinal canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T5354241 — 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: Feferman–Schütte ordinal
Context triple: [Solomon Feferman, notableIdea, Feferman–Schütte ordinal]

A. 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.
B. 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.
C. 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.
D. Fraenkel
Fraenkel is a surname most prominently associated with Abraham Fraenkel, a German-Israeli mathematician known for his foundational work in axiomatic set theory.
E. von Neumann–Bernays–Gödel set theory
Von Neumann–Bernays–Gödel set theory is an axiomatic set theory extending Zermelo–Fraenkel set theory by formally distinguishing between sets and classes, widely used in foundational studies of mathematics.
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: Feferman–Schütte ordinal
Target entity description: 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.

A. 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.
B. 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.
C. 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.
D. Fraenkel
Fraenkel is a surname most prominently associated with Abraham Fraenkel, a German-Israeli mathematician known for his foundational work in axiomatic set theory.
E. von Neumann–Bernays–Gödel set theory
Von Neumann–Bernays–Gödel set theory is an axiomatic set theory extending Zermelo–Fraenkel set theory by formally distinguishing between sets and classes, widely used in foundational studies of mathematics.
F. None of above. chosen

Statements (44)

Predicate	Object
instanceOf	countable ordinal ⓘ large countable ordinal ⓘ ordinal number ⓘ proof-theoretic ordinal ⓘ
alsoKnownAs	Γ₀ NERFINISHED ⓘ
appearsInWorkOf	Kurt Schütte NERFINISHED ⓘ Solomon Feferman NERFINISHED ⓘ
associatedWithSystem	predicative analysis ⓘ predicative second-order arithmetic ⓘ
belongsTo	ordinal notation systems for predicative theories ⓘ
characterizedAs	the limit of the sequence φ_0(0), φ_1(0), φ_2(0), … in the Veblen hierarchy ⓘ the smallest ordinal α such that φ_α(0)=α in the Veblen hierarchy ⓘ
definedUsing	Veblen hierarchy NERFINISHED ⓘ
field	mathematical logic ⓘ proof theory ⓘ set theory ⓘ
greaterThan	all ordinals reachable by finite iteration of the Veblen function starting from 0 ⓘ ε₀ ⓘ
hasCofinality	ω ⓘ
hasProperty	closed under ordinal addition, multiplication, and exponentiation below it ⓘ first impredicative ordinal in many traditional analyses of predicativity ⓘ
isAdditivelyIndecomposable	true ⓘ
isCountable	true ⓘ
isEpsilonNumber	false ⓘ
isFirstFixedPointOf	Veblen function φ_α(0) ⓘ
isLimitOfIncreasingSequenceOfOrdinals	true ⓘ
isLimitOrdinal	true ⓘ
isMultiplicativelyIndecomposable	true ⓘ
isRecursiveOrdinal	true ⓘ
isStrictlyLessThan	Bachmann–Howard ordinal ⓘ
isWellOrdered	true ⓘ
lessThan	small Veblen ordinal NERFINISHED ⓘ
marksBoundaryOf	predicative mathematics ⓘ predicative provability strength ⓘ
namedAfter	Kurt Schütte NERFINISHED ⓘ Solomon Feferman NERFINISHED ⓘ
roleInProofTheory	ordinal of predicative analysis ⓘ proof-theoretic ordinal of predicative arithmetic and analysis ⓘ
symbol	Γ₀ NERFINISHED ⓘ
topicIn	predicativity debates in foundations of mathematics ⓘ proof-theoretic ordinal classification ⓘ
usedIn	foundations of mathematics ⓘ ordinal analysis ⓘ proof theory ⓘ

How these facts were elicited

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Solomon Feferman → notableIdea → Feferman–Schütte ordinal ⓘ

Veblen hierarchy → relatedTo → Feferman–Schütte ordinal ⓘ

this entity surface form: Feferman–Schütte ordinal Γ₀

Veblen hierarchy → canReach → Feferman–Schütte ordinal ⓘ