Ordinal number- The mathematical concept
Labels: EducationThis article is about the mathematical concept. For the series of words ("first", "second", "third", etc.), see Ordinal number (linguistics).
Representation of the ordinal numbers up to ωω. Each turn of the spiral represents one power of ω. In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. Ordinals are an extension of the natural numbers different from integers and from cardinals. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated. The finite ordinals (and the finite cardinals) are the natural numbers: 0, 1, 2, …, since any two total orderings of a finite set are order isomorphic. The least infinite ordinal is ω which is identified with the cardinal number \aleph_0. Beyond ω however (the transfinite case) ordinals draw a finer distinction than cardinals on account of their order information. Whereas there is only one countably infinite cardinal, namely \aleph_0 itself, there are uncountably many countably infinite ordinals, namely ω, ω + 1, ω + 2, …, ω·2, ω·2 + 1, …, ω2, …, ω3, …, ωω, …, ωωω, …, ε0, …. Here addition and multiplication are not commutative: in particular 1 + ω is ω rather than ω + 1, while 2·ω is ω rather than ω·2. The set of all countable ordinals constitutes the first uncountable ordinal ω1 which is identified with the cardinal \aleph_1 (next cardinal after \aleph_0). Well-ordered cardinals are identified with their initial ordinals, i.e. the smallest ordinal of that cardinality. The cardinality of an ordinal defines a many to one association from ordinals to cardinals...
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Representation of the ordinal numbers up to ωω. Each turn of the spiral represents one power of ω. In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. Ordinals are an extension of the natural numbers different from integers and from cardinals. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated. The finite ordinals (and the finite cardinals) are the natural numbers: 0, 1, 2, …, since any two total orderings of a finite set are order isomorphic. The least infinite ordinal is ω which is identified with the cardinal number \aleph_0. Beyond ω however (the transfinite case) ordinals draw a finer distinction than cardinals on account of their order information. Whereas there is only one countably infinite cardinal, namely \aleph_0 itself, there are uncountably many countably infinite ordinals, namely ω, ω + 1, ω + 2, …, ω·2, ω·2 + 1, …, ω2, …, ω3, …, ωω, …, ωωω, …, ε0, …. Here addition and multiplication are not commutative: in particular 1 + ω is ω rather than ω + 1, while 2·ω is ω rather than ω·2. The set of all countable ordinals constitutes the first uncountable ordinal ω1 which is identified with the cardinal \aleph_1 (next cardinal after \aleph_0). Well-ordered cardinals are identified with their initial ordinals, i.e. the smallest ordinal of that cardinality. The cardinality of an ordinal defines a many to one association from ordinals to cardinals...
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