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.
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 NERFINISHED ⓘ |
| logicalStrength | its totality for all countable ordinals is not provable in very weak arithmetic theories ⓘ |
| namedAfter | Georg Cantor NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Veblen hierarchy NERFINISHED ⓘ 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 NERFINISHED ⓘ |
| usesSymbol | ω ⓘ |
| zeroCase | 0 is represented by the empty sum ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.