Førstetellbart

Førstetellbart 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 set is infinite, but its infinity is of the smallest possible size, often referred to as countably infinite.

A set is considered førstetellbart if there exists a bijection (a one-to-one and onto function) between the set and the set of natural numbers {0, 1, 2, 3, ...} or {1, 2, 3, ...}. This correspondence allows us to list the elements of the set in an ordered sequence, even though the sequence is infinitely long. For example, the set of all integers is førstetellbart because we can arrange them as 0, 1, -1, 2, -2, 3, -3, and so on.

The set of rational numbers is also förstetellbart. Despite appearing denser than integers, they can still be systematically enumerated. This property is crucial in understanding the structure of different sizes of infinity. Sets that are not förstetellbart are called uncountably infinite. The set of real numbers, for instance, is uncountably infinite, meaning there is no way to list all its elements in an infinite sequence.

The concept of førstetellbart was rigorously developed by Georg Cantor in the late 19th century. His work on set theory, including the distinction between different cardinalities of infinity, revolutionized mathematics. The term itself is a Norwegian translation of the German "abzählbar unendlich" or the English "countably infinite."