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Any ordinal number can be made into a topological space by endowing it with the order topology; this topology is discrete if and only if the ordinal is a countable cardinal, i.e. at most ω. A subset of ω + 1 is open in the order topology if and only if either it is cofinite or it does not contain ω as an element.
Von Neumann cardinal assignment implies that the cardinal number of a finite set is the common ordinal number of all possible well-orderings of that set, and cardinal and ordinal arithmetic (addition, multiplication, power, proper subtraction) then give the same answers for finite numbers. However, they differ for infinite numbers.
Cardinal versus ordinal numbers. In linguistics, ordinal numerals or ordinal number words are words representing position or rank in a sequential order; the order may be of size, importance, chronology, and so on (e.g., "third", "tertiary"). They differ from cardinal numerals, which represent quantity (e.g., "three") and other types of numerals.
For each ordinal, + is the least cardinal number greater than . The cardinality of the natural numbers is denoted aleph-null ( ℵ 0 {\displaystyle \aleph _{0}} ), while the cardinality of the real numbers is denoted by " c {\displaystyle {\mathfrak {c}}} " (a lowercase fraktur script "c"), and is also referred to as the cardinality of the ...
In linguistics, and more precisely in traditional grammar, a cardinal numeral (or cardinal number word) is a part of speech used to count. Examples in English are the words one, two, three, and the compounds three hundred [and] forty-two and nine hundred [and] sixty. Cardinal numerals are classified as definite, and are related to ordinal ...
The least ordinal of cardinality ℵ 0 (that is, the initial ordinal of ℵ 0) is ω but many well-ordered sets with cardinal number ℵ 0 have an ordinal number greater than ω. For finite well-ordered sets, there is a one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by the same natural ...
The von Neumann cardinal assignment is a cardinal assignment that uses ordinal numbers. For a well-orderable set U, we define its cardinal number to be the smallest ordinal number equinumerous to U, using the von Neumann definition of an ordinal number. More precisely: where ON is the class of ordinals. This ordinal is also called the initial ...
The cardinal exponentiation A B is defined to be the cardinal number of the set of all functions B → A, while the ordinal exponentiation α β only contains the functions β → α with finite support, typically a set of much smaller cardinality.