Alefszámok

Alefszámok, often translated as aleph numbers, are a sequence of numbers used in set theory to represent the cardinalities of infinite sets. They are denoted by the Hebrew letter aleph ($\aleph$) followed by an ordinal number subscript. The first aleph number, $\aleph_0$, represents the cardinality of the set of natural numbers, which is the smallest infinite cardinality. This means that any set whose elements can be put into a one-to-one correspondence with the natural numbers has cardinality $\aleph_0$.

The next aleph number, $\aleph_1$, represents the smallest cardinality strictly greater than $\aleph_0$. The continuum hypothesis states that the cardinality of the set of real numbers is equal to $\aleph_1$. However, the continuum hypothesis has been proven to be independent of the standard axioms of set theory (ZFC), meaning it can neither be proven nor disproven within that system.

Subsequent aleph numbers, such as $\aleph_2$, $\aleph_3$, and so on, represent increasingly larger infinite cardinalities. The sequence of aleph numbers is constructed such that $\aleph_{\alpha+1}$ is the smallest cardinality strictly greater than $\aleph_\alpha$ for any ordinal $\alpha$. For limit ordinals $\lambda$, $\aleph_\lambda$ is defined as the supremum of the cardinalities $\{\aleph_\beta \mid \beta < \lambda\}$. Aleph numbers are fundamental to understanding the hierarchy of infinite sizes and the structure of infinite sets in mathematics.