alefnul

Aleph-null, also written aleph-null or aleph-zero, is the smallest infinite cardinal number in the standard hierarchy of set theory. It is denoted by the symbol ℵ0 and is defined as the cardinality of the set of natural numbers N = {0,1,2,...}. More generally, any countably infinite set has cardinality ℵ0: there exists a bijection between the set and N.

Examples of countably infinite sets include the natural numbers N, the integers Z, and the rational numbers Q. Although these sets are infinite, they can be put into a sequential order, which is the defining property of countability.

A fundamental consequence is that the set of real numbers R is not countable. Cantor’s diagonal argument shows that |R| = c = 2^ℵ0, which is strictly larger than ℵ0. Thus aleph-null is the starting point for the infinite cardinal hierarchy, and it is the size of any countably infinite set.

ℵ0 is the first in the sequence of aleph numbers: ℵ0 < ℵ1 < ℵ2 < …. Under the axiom of choice, these form a well-ordered progression of infinite cardinals. The continuum hypothesis questions whether c equals ℵ1; this question is independent of ZFC, meaning both the hypothesis and its negation are consistent with ZFC relative to other assumptions.

In summary, aleph-null provides a formal measure for the size of basic infinite sets and serves as the foundational element in the study of infinite cardinalities.