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Cantor's work with derived sets and ordinal numbers led to the Cantor-Bendixson theorem. [14] Using successors, limits, and cardinality, Cantor generated an unbounded sequence of ordinal numbers and number classes. [15] The (α + 1)-th number class is the set of ordinals whose predecessors form a set of the same cardinality as the α-th
Ordinal indicator – Character(s) following an ordinal number (used when writing ordinal numbers, such as a super-script) Ordinal number – Generalization of "n-th" to infinite cases (the related, but more formal and abstract, usage in mathematics) Ordinal data, in statistics; Ordinal date – Date written as number of days since first day of ...
Transfinite numbers: Numbers that are greater than any natural number. Ordinal numbers: Finite and infinite numbers used to describe the order type of well-ordered sets. Cardinal numbers: Finite and infinite numbers used to describe the cardinalities of sets.
If β = ω λ m + ω μ n + smaller terms, then β = (ω μ n + smaller terms)(ω λ−μ + 1)m is a product of a smaller ordinal and a prime and a natural number m. Repeating this and factorizing the natural numbers into primes gives the prime factorization of β. So the factorization of the Cantor normal form ordinal ω α 1 n 1 + ⋯ + ω α ...
The ordinal catgegory are based on ordinal numbers such as the English first, second, third, which specify position of items in a sequence. In Latin and Greek, the ordinal forms are also used for fractions for amounts higher than 2; only the fraction 1 / 2 has special forms.
Numbers may either precede or follow their noun (see Latin word order). Most numbers are invariable and do not change their endings: regnāvit Ancus annōs quattuor et vīgintī (Livy) [1] 'Ancus reigned for 24 years' However, the numbers 1, 2, 3, and 200, 300, etc. change their endings for gender and grammatical case.
The standard definition of ordinal exponentiation with base α is: =, =, when has an immediate predecessor . = {< <}, whenever is a limit ordinal. From this definition, it follows that for any fixed ordinal α > 1, the mapping is a normal function, so it has arbitrarily large fixed points by the fixed-point lemma for normal functions.
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