Cantor normal form
E886917
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Cantor normal form canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10829094 — 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: Cantor normal form Context triple: [Veblen hierarchy, generalizes, Cantor normal form]
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A.
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.
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B.
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.
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C.
Knuth’s up-arrow notation
Knuth’s up-arrow notation is a mathematical notation introduced by Donald Knuth to concisely represent very large integers using iterated exponentiation and its higher-order generalizations.
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D.
Kleene’s normal form theorem
Kleene’s normal form theorem is a fundamental result in computability theory that characterizes all partial recursive (effectively computable) functions using a universal primitive recursive function and the μ-operator.
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E.
Cantor’s theorem
Cantor’s theorem is a fundamental result in set theory stating that the power set of any set has a strictly greater cardinality than the set itself, implying there is no largest infinity.
- 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: Cantor normal form Target entity description: 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.
-
A.
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.
-
B.
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.
-
C.
Knuth’s up-arrow notation
Knuth’s up-arrow notation is a mathematical notation introduced by Donald Knuth to concisely represent very large integers using iterated exponentiation and its higher-order generalizations.
-
D.
Kleene’s normal form theorem
Kleene’s normal form theorem is a fundamental result in computability theory that characterizes all partial recursive (effectively computable) functions using a universal primitive recursive function and the μ-operator.
-
E.
Cantor’s theorem
Cantor’s theorem is a fundamental result in set theory stating that the power set of any set has a strictly greater cardinality than the set itself, implying there is no largest infinity.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
canonical form
ⓘ
mathematical concept ⓘ ordinal notation ⓘ |
| appliesTo | ordinal numbers ⓘ |
| baseOrdinal | ω ⓘ |
| canonicalFor | well-ordered sets up to order isomorphism ⓘ |
| coefficientType | natural numbers ⓘ |
| constraintOnCoefficients | ci ∈ ℕ and ci ≥ 1 ⓘ |
| constraintOnExponents | α1 > α2 > … > αn ≥ 0 ⓘ |
| domain | countable ordinals ⓘ |
| enables | lexicographic comparison of ordinals ⓘ |
| exponentType | ordinals ⓘ |
| field |
mathematical logic
ⓘ
ordinal arithmetic ⓘ set theory ⓘ |
| generalizes | base-ω expansion of natural numbers ⓘ |
| historicalPeriod | late 19th century ⓘ |
| includesCase | zero ordinal has empty sum representation ⓘ |
| introducedBy | Georg Cantor ⓘ |
| logicalStrength | its totality for all countable ordinals is not provable in very weak arithmetic theories ⓘ |
| namedAfter | Georg Cantor ⓘ |
| notationExample | ω^α1·c1 + ω^α2·c2 + … + ω^αn·cn ⓘ |
| property |
all coefficients are positive integers
ⓘ
exponents form a strictly decreasing sequence of ordinals ⓘ only finitely many nonzero terms ⓘ |
| relatedConcept |
Cantor–Bendixson rank
ⓘ
Veblen hierarchy ⓘ epsilon numbers ⓘ ordinal collapsing function ⓘ |
| representationType | finite sum ⓘ |
| requires | strictly decreasing exponents ⓘ |
| standardFormFor | ordinal arithmetic ⓘ |
| uniqueness | every nonzero ordinal has a unique Cantor normal form ⓘ |
| usedFor |
comparing ordinals
ⓘ
defining ordinal addition ⓘ defining ordinal exponentiation ⓘ defining ordinal multiplication ⓘ proofs about ordinal recursion ⓘ proofs about transfinite induction ⓘ |
| usedIn |
descriptive set theory
ⓘ
ordinal analysis ⓘ proof theory ⓘ |
| usedToDefine | Cantor–Bendixson normal form for closed sets of reals ⓘ |
| usesSymbol | ω ⓘ |
| zeroCase | 0 is represented by the empty sum ⓘ |
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: Cantor normal form Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.