uppräkneligt

Uppräkneligt is a mathematical term used primarily in Swedish, corresponding to the English word countable. A set is uppräkneligt if it is finite or if there exists a bijection between the set and a subset of the natural numbers. Equivalently, there exists an injection from the set into the natural numbers or a surjection from the natural numbers onto the set. Finite sets are always uppräknelige, and in general a set is considered uppräkneligt if its elements can be listed in a sequence without omitting any element.

Examples include the set of natural numbers N, the set of integers Z, the set of rational numbers Q, and the set of algebraic numbers. More broadly, any finite string or finite combination formed from a finite alphabet is uppräkneligt, and the set of all such finite objects is countable.

A crucial distinction is between countable and uncountable sets. The set of real numbers R is not uppräkneligt, as Cantor's diagonal argument shows that the reals are uncountable. Similarly, the power set of N, P(N), is uncountable. These results lead to the formal separation between countable (uppräkneligt) and uncountable sets in set theory.

Notes on terminology: denumerabel is often used to denote countably infinite (i.e., in bijection with N). Uppräkneligt generally covers finite or countably infinite sets, with denumerabel serving as a more precise subtype. In practice, the terms are closely related and sometimes used interchangeably in non-technical writing.