ALEPH NUMBER Meaning and
Definition
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The term "aleph number" refers to a mathematical concept representing the cardinality or size of infinite sets. In set theory, it is used to measure the "number" of elements or members in a set, even if that set is infinite in size. Specifically, aleph numbers were introduced by the mathematician Georg Cantor in the late 19th century to study the different sizes of infinite sets.
An aleph number, denoted as \(\aleph\), is defined as the smallest possible cardinality of an infinite set. The smallest aleph number, \(\aleph_0\), called aleph null or aleph zero, represents the cardinality of the set of natural numbers (0, 1, 2, 3, ...). It signifies the countable infinity.
Cantor also formulated a hierarchy of aleph numbers, known as the aleph sequence. By taking the power set of each aleph number (i.e., considering the set of all subsets of a given set), Cantor showed that there is always a larger aleph number than any previously defined one. This led to the concept of uncountable infinities, which he denoted as \(\aleph_1\), \(\aleph_2\), \(\aleph_3\), and so on.
The notion of aleph numbers plays a fundamental role in the realm of transfinite arithmetic and set theory. They help mathematicians analyze and compare the sizes of infinite sets, opening up avenues for understanding the rich and intricate nature of infinite mathematics.
Common Misspellings for ALEPH NUMBER
- zleph number
- sleph number
- wleph number
- qleph number
- akeph number
- apeph number
- aoeph number
- alwph number
- alsph number
- aldph number
- alrph number
- al4ph number
- al3ph number
- aleoh number
- alelh number
- ale-h number
- ale0h number
- alepg number
- alepb number
- alepn number
Etymology of ALEPH NUMBER
The term "aleph number" is derived from the Hebrew letter "aleph" (א), which represents the first letter of the Hebrew alphabet. It was introduced into mathematics by the German mathematician Georg Cantor in the late 19th century to describe different sizes of infinity. Cantor named the first countable infinite cardinal number as "aleph-null" (ℵ₀) with the use of the Hebrew letter "aleph" followed by the subscript representing zero to symbolize the smallest infinite cardinality. From there, Cantor continued to define larger infinite cardinal numbers using the aleph symbol, such as ℵ₁, ℵ₂, and so on, each representing a higher cardinality. Hence, the term "aleph number" refers to the infinite cardinal numbers introduced by Cantor.
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