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In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, n th, etc.) aimed to extend enumeration to infinite sets. [ 1 ] A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used.
Also, is the smallest uncountable ordinal (to see that it exists, consider the set of equivalence classes of well-orderings of the natural numbers; each such well-ordering defines a countable ordinal, and is the order type of that set), is the smallest ordinal whose cardinality is greater than , and so on, and is the limit of for natural ...
Download as PDF; Printable version; In other projects ... Pages in category "Ordinal numbers" The following 51 pages are in this category, out of 51 total.
In other words, every ordinal number α can be uniquely written as + + +, where k is a natural number, and … are ordinal numbers. Another variation of the Cantor normal form is the "base δ expansion", where ω is replaced by any ordinal δ > 1 , and the numbers c i are nonzero ordinals less than δ .
Any finite natural number can be used in at least two ways: as an ordinal and as a cardinal. Cardinal numbers specify the size of sets (e.g., a bag of five marbles), whereas ordinal numbers specify the order of a member within an ordered set [9] (e.g., "the third man from the left" or "the twenty-seventh day of January").
Since the epsilon numbers are an unbounded subclass of the ordinal numbers, they are enumerated using the ordinal numbers themselves. For any ordinal number , is the least epsilon number (fixed point of the exponential map) not already in the set {<}. It might appear that this is the non-constructive equivalent of the constructive definition ...
In computer programming, an ordinal data type is a data type with the property that its values can be counted. That is, the values can be put in a one-to-one correspondence with the positive integers .
For example, the smallest limit ordinal is ω, the smallest ordinal greater than every natural number. This is a limit ordinal because for any smaller ordinal (i.e., for any natural number) n we can find another natural number larger than it (e.g. n+1), but still less than ω. The next-smallest limit ordinal is ω+ω.
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