alefszámokkal

Alefszámmal, also known as aleph numbers, are a sequence of numbers that represent the cardinalities of infinite well-ordered sets. The first infinite cardinal number is aleph-null, denoted as $\aleph_0$, which is the cardinality of the set of natural numbers. This means that any set that can be put into a one-to-one correspondence with the natural numbers has cardinality $\aleph_0$. Examples include the set of integers and the set of rational numbers.

The next aleph number, aleph-one, denoted as $\aleph_1$, is the smallest cardinal number greater than $\aleph_0$. It represents the cardinality of the smallest set that has more elements than the set of natural numbers. The question of whether $\aleph_1$ is equal to the cardinality of the set of real numbers (continuum hypothesis) is a famous undecidable problem in set theory.

In general, the sequence of aleph numbers is defined recursively. For any ordinal number $\alpha$, the aleph number $\aleph_{\alpha+1}$ is defined as the smallest cardinal number strictly greater than $\aleph_{\alpha}$. For a limit ordinal $\lambda$, $\aleph_{\lambda}$ is defined as the supremum of all $\aleph_{\beta}$ for $\beta < \lambda$. This construction allows for the enumeration of all infinite cardinal numbers. The hierarchy of aleph numbers continues indefinitely: $\aleph_0, \aleph_1, \aleph_2, \dots, \aleph_\omega, \aleph_{\omega+1}, \dots$. They are fundamental to understanding the sizes of different infinite sets within the framework of Zermelo-Fraenkel set theory.