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Ordinal numbers may be written in English with numerals and letter suffixes: 1st, 2nd or 2d, 3rd or 3d, ... Deutero-Isaiah. [3] Numbers beyond three are rare; those ...
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.
Computable ordinals (or recursive ordinals) are certain countable ordinals: loosely speaking those represented by a computable function.There are several equivalent definitions of this: the simplest is to say that a computable ordinal is the order-type of some recursive (i.e., computable) well-ordering of the natural numbers; so, essentially, an ordinal is recursive when we can present the set ...
The cardinality of any infinite ordinal number is an aleph number. Every aleph is the cardinality of some ordinal. The least of these is its initial ordinal. Any set whose cardinality is an aleph is equinumerous with an ordinal and is thus well-orderable. Each finite set is well-orderable, but does not have an aleph as its cardinality.
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]
An ordinal is stable if is a -elementary-substructure of , denoted . [9] These are some of the largest named nonrecursive ordinals appearing in a model-theoretic context, for instance greater than {:} for any computably axiomatizable theory . [10]
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 .
It is the proof-theoretic ordinal of several mathematical theories, such as arithmetical transfinite recursion. It is named after Solomon Feferman and Kurt Schütte, the former of whom suggested the name Γ 0. [1] There is no standard notation for ordinals beyond the Feferman–Schütte ordinal.