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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 . Infinitesimals : These are smaller than any positive real number, but are nonetheless greater than zero.
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
The ordinal numbers are difficult to reconstruct due to their significant variation in the daughter languages. The following reconstructions are tentative: [ 20 ] "first" is formed with * pr̥h₃- (related to some adverbs meaning "forth, forward, front" and to the particle * prō "forth", thus originally meaning "foremost" or similar) plus ...
The ordinal category 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.
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 ...
So the factorization of the Cantor normal form ordinal ω α 1 n 1 + ⋯ + ω α k n k (with α 1 > ⋯ > α k) into a minimal product of infinite primes and natural numbers is (ω ω β 1 ⋯ ω ω β m)n k (ω α k−1 −α k + 1)n k−1 ⋯ (ω α 1 −α 2 + 1)n 1. where each n i should be replaced by its factorization into a non ...