Aleph-nought, aleph-zero, or aleph-null, the smallest infinite cardinal number. In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets.

Mar 7, 2025 · aleph-null (ℵ 0), in mathematics, the cardinality of the infinite set of natural numbers {1, 2, 3, …}. The cardinality, or cardinal number, of a set is the number of elements of a set.

In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. They were introduced by the mathematician Georg Cantor [1] and are named after the symbol he …

Jan 2, 2021 · As symbols, alephs were introduced by G. Cantor to denote the cardinal numbers (i.e., the cardinality) of infinite well-ordered sets. Each cardinal number is some aleph (a consequence of the axiom of choice). However, many theorems about alephs are demonstrated without recourse to the axiom of choice.

$\aleph_3$ is the cardinality of the set of all ordinal numbers of cardinality $\le\aleph_2$. ${}\qquad\vdots$ $\aleph_\omega$ is the cardinality of the set of all ordinal numbers of cardinality no bigger than some $\aleph_n$ for finite $n$.

Aleph' is the first letter of the Hebrew alphabet. Written ℵ {\displaystyle \aleph} . The so-called "variable" is used in mathematics subject of set theory to denote a type of cardinality (set size).

Nov 11, 2021 · The aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to …

The cardinal number $\aleph_1$ is defined as the cardinality of the set of all countable ordinals, i.e. ordinals of cardinality $\le\aleph_0$. If you believe that that set of countable ordinals exists, then you've got it.

Oct 11, 2021 · Aleph ($\aleph$) is the symbol used in mathematics to indicate the cardinality of the numerable. We know that an infinite set has cardinality $\aleph_0$ if there exists a bijection that relates it biunivocally to the set $ \mathbb {N}$ to $ \mathbb {N}$ of the natural numbers.

Mar 5, 2020 · The set theory symbol (aleph) for the cardinal number of a well-orderable infinite set.