Abzählbaren

Abzählbaren is a term used in mathematics, specifically in set theory, to describe a set whose elements can be put into a one-to-one correspondence with the natural numbers. This means that the elements of an abzählbaren set can be listed in a sequence, even if the sequence is infinite. Sets that are abzählbaren are also called countable sets or denumerable sets.

There are two types of abzählbaren sets: finite and countably infinite. A finite abzählbaren set is simply a set with a finite number of elements. A countably infinite abzählbaren set has an infinite number of elements, but its elements can still be enumerated. This is a crucial distinction from uncountable sets, whose elements cannot be listed in such a way.

Examples of abzählbaren sets include the set of natural numbers itself {1, 2, 3, ...}, the set of integers {..., -2, -1, 0, 1, 2, ...}, and the set of rational numbers. Despite the apparent density of rational numbers, they can be systematically listed. The concept of abzählbaren is fundamental to understanding different sizes of infinity and plays a key role in areas like computability theory and the foundations of mathematics.