Search results
Results From The WOW.Com Content Network
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.
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 ...
Such a number is algebraic and can be expressed as the sum of a rational number and the square root of a rational number. Constructible number: A number representing a length that can be constructed using a compass and straightedge. Constructible numbers form a subfield of the field of algebraic numbers, and include the quadratic surds.
Pages in category "Ordinal numbers" The following 51 pages are in this category, out of 51 total. This list may not reflect recent changes. ...
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation.Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set that represents the result of the operation or by using transfinite recursion.
An initial segment of the von Neumann universe. Ordinal multiplication is reversed from our usual convention; see Ordinal arithmetic. The cumulative hierarchy is a collection of sets V α indexed by the class of ordinal numbers; in particular, V α is the set of all sets having ranks less than α. Thus there is one set V α for each ordinal ...
The set of ordinal numbers is the set of all order types of well-orderings. This does not work in ZFC , because the equivalence classes are too large. It would be formally possible to use Scott's trick to define the ordinals in essentially the same way, but a device of von Neumann is more commonly used.
Every well-ordered set is order-equivalent to exactly one ordinal number, by definition. The ordinal numbers are taken to be the canonical representatives of their classes, and so the order type of a well-ordered set is usually identified with the corresponding ordinal. Order types thus often take the form of arithmetic expressions of ordinals.