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Greek letters are used in mathematics, science, engineering, and other areas where mathematical notation is used as symbols for constants, special functions, and also conventionally for variables representing certain quantities. In these contexts, the capital letters and the small letters represent distinct and unrelated entities.
In Archaic and early Classical times, the Greek alphabet existed in many local variants, but, by the end of the 4th century BC, the Ionic-based Euclidean alphabet, with 24 letters, ordered from alpha to omega, had become standard throughout the Greek-speaking world [6] and is the version that is still used for Greek writing today.
The definition of addition α + β can also be given by transfinite recursion on β. When the right addend β = 0, ordinary addition gives α + 0 = α for any α. For β > 0, the value of α + β is the smallest ordinal strictly greater than the sum of α and δ for all δ < β. Writing the successor and limit ordinals cases separately: α + 0 = α
For example, the ordinal 42 is generally identified as the set {0, 1, 2, ..., 41}. Conversely, any set S of ordinals that is downward closed — meaning that for any ordinal α in S and any ordinal β < α, β is also in S — is (or can be identified with) an ordinal. This definition of ordinals in terms of sets allows for infinite ordinals.
The definition of a Greek letter for this list is a character encoded in the Unicode standard that a has script property of "Greek" and the general category of "Letter". An overview of the distribution of Greek letters is given in Greek script in Unicode .
The definition of ℵ 1 implies (in ZF, Zermelo–Fraenkel set theory without the axiom of choice) that no cardinal number is between ℵ 0 and ℵ 1. If the axiom of choice is used, it can be further proved that the class of cardinal numbers is totally ordered , and thus ℵ 1 is the second-smallest infinite cardinal number.
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The rationale behind this is that it enables design and usage of special mathematical characters that include all necessary properties to differentiate from other alphanumerics, e.g. in mathematics an italic "𝐴" can have a different meaning from a roman letter "A".