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In mathematics, transfinite numbers or infinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers. These include the transfinite cardinals, which are cardinal numbers used to quantify the size of infinite sets, and the transfinite ordinals, which are ordinal numbers used to provide an ordering of infinite sets.
The transfinite ordinal numbers, which first appeared in 1883, [8] originated in Cantor's work with derived sets. If P is a set of real numbers, the derived set P ′ is the set of limit points of P. In 1872, Cantor generated the sets P (n) by applying the derived set operation n times to P.
Transfinite induction requires proving a base case (used for 0), a successor case (used for those ordinals which have a predecessor), and a limit case (used for ordinals which don't have a predecessor). Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers.
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.
Since the epsilon numbers are an unbounded subclass of the ordinal numbers, they are enumerated using the ordinal numbers themselves. For any ordinal number , is the least epsilon number (fixed point of the exponential map) not already in the set {<}. It might appear that this is the non-constructive equivalent of the constructive definition ...
ℵ 1 is, by definition, the cardinality of the set of all countable ordinal numbers. This set is denoted by ω 1 (or sometimes Ω). The set ω 1 is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore, ℵ 1 is distinct from ℵ 0.
Continuing to perform transfinite induction beyond S ω produces more ordinal numbers α, each represented as the largest surreal number having birthday α. (This is essentially a definition of the ordinal numbers resulting from transfinite induction.) The first such ordinal is ω + 1 = { ω | }.
Beth numbers are defined by transfinite recursion: =, + =, = {: <}, where is an ordinal and is a limit ordinal. [1]The cardinal = is the cardinality of any countably infinite set such as the set of natural numbers, so that = | |.