felsorolhatóság

Felsorolhatóság is a concept in theoretical computer science and mathematics related to sets. A set is considered felsorolható (enumerable or countable) if its elements can be put into a one-to-one correspondence with the set of natural numbers. This means that even if the set is infinite, its elements can be listed in some order, perhaps with repetitions, such that every element appears in the list eventually.

Formally, a set A is felsorolható if there exists a surjective function from the set of natural numbers (N) to A. Alternatively, a set is felsorolható if it is finite or countably infinite. Countably infinite sets are those that can be put into a one-to-one correspondence with the natural numbers themselves.

Examples of felsorolható sets include the set of integers (Z), the set of rational numbers (Q), and the set of all finite strings over a given alphabet. The set of real numbers (R), however, is not felsorolható; it is uncountably infinite.

The concept of felsorolhatóság is fundamental in computability theory. A set is recursively enumerable (or computably enumerable) if it is the set of all inputs for which a particular algorithm halts. This is a slightly stronger condition than just being felsorolható, as it implies that the elements can be listed by an algorithm. The Church-Turing thesis suggests that anything that can be intuitively computed can be computed by a Turing machine, linking felsorolhatóság to the limits of computation.