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

2025年3月7日 · Aleph-null (ℵ0), 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. For example, the number 3 is the cardinality of the set {1, 2, …

Aleph null (also aleph naught or aleph 0) is the smallest infinite number. It is the cardinality (size) of the set of natural numbers (there are aleph null natural numbers). Georg Cantor invented and named the concept. The symbol for aleph null is .[1] It is the first infinite number in a series of infinite cardinal numbers.

2019年1月2日 · If the infinite cardinals aleph-null, aleph-two, etc. continue indefinitely, is there any meaning in the idea of aleph-aleph-null? In this post people are talk about the difference between cardinal numbers and how ℵℵ0 ℵ ℵ 0 should instead be ℵω ℵ ω. The responses to the post then go on to talk about ℵω+1 ℵ ω + 1, ℵω+2 ℵ ω + 2, and so on.

2022年6月23日 · Let ω ω denote the minimally inductive set. where ℵ ℵ denotes the aleph mapping. For all n ∈ ω n ∈ ω, n ∉ N′ n ∉ N ′ by the definition of the class of infinite cardinals. Therefore, ω ≤ℵ0 ω ≤ ℵ 0. . Moreover, ω ∈N′ ω ∈ N ′ by Minimally Inductive Set is Infinite Cardinal. Therefore, ℵx = ω ℵ x = ω for some ordinal x x.

2021年11月11日 · 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 ( ).

2025年1月31日 · aleph - null (uncountable) (set theory) The first of the transfinite cardinal numbers; corresponds to the number of natural numbers.

4 天之前 · The symbol is often pronounced "aleph-null" rather than "aleph-zero," probably because Null is the word for "zero" in Georg Cantor's native language of German. It is sometimes also pronounced "aleph-zero" or "aleph-naught," the latter of which is also spelled "aleph-nought." The algebraic numbers also have cardinality .

2017年12月5日 · The right definition of the factorial is as the size of the corresponding group of permutations: remember that in the finite case, n! n! is the number of permutations of an n n -element set, and this generalizes immediately to the κ κ -case.

Alef-noll. Alef-noll , ℵ 0 {\displaystyle \aleph _{0}} , är kardinaltalet för alla uppräkneligt oändliga mängder . Exempel på sådana mängder är de naturliga talen och heltalen .