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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.
Transfinite number, a number larger than all finite numbers, yet not absolutely infinite; Transfinite induction, an extension of mathematical induction to well-ordered sets Transfinite recursion; Transfinite arithmetic, the generalization of elementary arithmetic to infinite quantities; Transfinite interpolation, a method in numerical analysis ...
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 = | |.
Transfinite numbers: Numbers that are greater than any natural number. Ordinal numbers: Finite and infinite numbers used to describe the order type of well-ordered sets. Cardinal numbers: Finite and infinite numbers used to describe the cardinalities of sets.
A list of articles about numbers (not about numerals). Topics include powers of ten, notable integers, prime and cardinal numbers, and the myriad system.
The absolute infinite (symbol: Ω), in context often called "absolute", is an extension of the idea of infinity proposed by mathematician Georg Cantor. It can be thought of as a number that is bigger than any other conceivable or inconceivable quantity, either finite or transfinite .
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.
In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map. Consequently, they are not reachable from 0 via a finite series of applications of the chosen exponential map and of "weaker" operations like addition and multiplication.