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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.
The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BC. [2] [3] It was derived from the earlier Phoenician alphabet, [4] and is the earliest known alphabetic script to have developed distinct letters for consonants as well as vowels. [5]
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 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
Omega (US: / oʊ ˈ m eɪ ɡ ə,-ˈ m ɛ ɡ ə,-ˈ m iː ɡ ə /, UK: / ˈ oʊ m ɪ ɡ ə /; [1] uppercase Ω, lowercase ω; Ancient Greek ὦ, later ὦ μέγα, Modern Greek ωμέγα) is the twenty-fourth and last letter in the Greek alphabet.
Alpha / ˈ æ l f ə / [1] (uppercase Α, lowercase α) [a] is the first letter of the Greek alphabet. In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician letter aleph, which is the West Semitic word for "ox". [2] Letters that arose from alpha include the Latin letter A and the Cyrillic letter А.
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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.