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To define this set, he defined the transfinite ordinal numbers and transformed the infinite indices into ordinals by replacing ∞ with ω, the first transfinite ordinal number. Cantor called the set of finite ordinals the first number class. The second number class is the set of ordinals whose predecessors form a countably infinite set.
Ordinal numbers may be written in English with numerals and letter suffixes: 1st, 2nd or 2d, 3rd or 3d, 4th, 11th, 21st, 101st, 477th, etc., with the suffix acting as an ordinal indicator. Written dates often omit the suffix, although it is nevertheless pronounced.
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.
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 .
Ordinal indicators are sometimes written as superscripts (1 st, 2 nd, 3 rd, 4 th, rather than 1st, 2nd, 3rd, 4th), although many English-language style guides recommend against this use. [4] Romance languages use a similar convention, such as 1 er or 2 e in French, or 4ª and 4º in Galician and Italian, or 4.ª and 4.º in Portuguese and Spanish.
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