unzählbare

Unzählbare is the inflected form of the German adjective unzählbar, meaning "not countable" or "innumerable." In everyday language it is used to describe things that are so numerous they cannot be counted, for example unzählbare Sterne or unzählbare Möglichkeiten. In mathematical contexts the term is closely aligned with the concept of uncountable, indicating sets whose elements cannot be put into a one-to-one correspondence with the natural numbers.

In German mathematics, abzählbar (countable) refers to sets that are finite or countably infinite, meaning they can be listed in a sequence like a1, a2, a3, …. Unzählbar (uncountable) characterizes sets that cannot be enumerated in such a way; there is no bijection between the set and the natural numbers. A canonical example is the set of real numbers, which Cantor showed to be unzählbar using his diagonal argument. Related unzählbare sets include the set of irrational numbers and the power set of the natural numbers, both of which are unzählbar but have larger cardinalities than the set of natural numbers.

The distinction between abzählbar and unzählbar underpins important results in analysis and set theory, such as the existence of uncountable sets with measure zero or the formulation of the continuum hypothesis, which concerns the size of the real numbers relative to the natural numbers. The term also appears in discussions of cardinal arithmetic and the structure of different infinities, highlighting the difference between quantities that can be enumerated and those that cannot.

In broader usage, unzählbar conveys the sense of something so numerous that counting is impractical or impossible, reinforcing the contrast between finite, countable, and uncountable collections.