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.

The ℵ ℵ numbers tell us how many people are in the queue, not how long it is. For finite queues the two notions coincide, but for infinite queues they do not. It is clear that the queue described by N N, and the queue described by adding one more after that are both countable.

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 …

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 denote them, the Hebrew letter aleph ( ℵ {\displaystyle \,\aleph \,} ).

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). Aleph's come in different relative sizes, each of which is denoted by a subscript...

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.

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.

Apr 8, 2025 · The set theory symbol (aleph) for the cardinal number of a well-orderable infinite set.

Apr 18, 2020 · $\aleph_0$ is the smallest infinite cardinal, i.e. the cardinality of $\mathbb{N}$. Under the von Neumann cardinal assignment, $\aleph_0$ is defined as $\omega_0$. However, using the symbol $\omega_0$ or $\aleph_0$ indicates whether we are treating it as an ordinal or as a cardinal, respectively.