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ℵ 0 (aleph-nought, aleph-zero, or aleph-null) is the cardinality of the set of all natural numbers, and is an infinite cardinal.The set of all finite ordinals, called ω or ω 0 (where ω is the lowercase Greek letter omega), also has cardinality ℵ 0.
n-superstrong (n≥2), n-almost huge, n-super almost huge, n-huge, n-superhuge cardinals (1-huge=huge, etc.) Wholeness axiom , rank-into-rank (Axioms I3, I2, I1, and I0) The following even stronger large cardinal properties are not consistent with the axiom of choice, but their existence has not yet been refuted in ZF alone (that is, without ...
: the first transfinite cardinal number. It is also the cardinality of the natural numbers. If the axiom of choice holds, the next higher cardinal number is aleph-one, . If not, there may be other cardinals which are incomparable with aleph-one and larger than aleph-null.
In gematria, aleph represents the number 1, and when used at the beginning of Hebrew years, it means 1000 (e.g. א'תשנ"ד in numbers would be the Hebrew date 1754, not to be confused with 1754 CE). Aleph, along with ayin, resh, he and heth, cannot receive a dagesh.
One-to-one correspondence between an infinite set and its proper subset. A different form of "infinity" is the ordinal and cardinal infinities of set theory—a system of transfinite numbers first developed by Georg Cantor. In this system, the first transfinite cardinal is aleph-null (ℵ 0), the cardinality of the set of natural numbers.
The cardinality of the set is the first uncountable cardinal number, . The ordinal ω 1 {\displaystyle \omega _{1}} is thus the initial ordinal of ℵ 1 {\displaystyle \aleph _{1}} . Under the continuum hypothesis , the cardinality of ω 1 {\displaystyle \omega _{1}} is ℶ 1 {\displaystyle \beth _{1}} , the same as that of R {\displaystyle ...
In set theory, , pronounced aleph-naught, aleph-zero, or aleph-null, is used to mark the cardinal number of an infinite countable set, such as , the set of all integers. More generally, the ℵ α {\displaystyle \aleph _{\alpha }} aleph number notation marks the ordered sequence of all distinct infinite cardinal numbers.
As is standard in set theory, we denote by the least infinite ordinal, which has cardinality ; it may be identified with the set of natural numbers.. A number of cardinal characteristics naturally arise as cardinal invariants for ideals which are closely connected with the structure of the reals, such as the ideal of Lebesgue null sets and the ideal of meagre sets.
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