Computably enumerable sets

Computably enumerable (c.e.) sets are collections of items that can be listed or generated by a computer program or algorithm, even if the process may never finish. This means there’s a systematic way to produce each element when asked, but the program might run indefinitely if the set is infinite. For example, the set of all prime numbers can be enumerated by an algorithm listing primes one after another. However, determining whether a specific number belongs to a c.e. set might be difficult or impossible in finite time, especially for infinite sets. These sets are fundamental in understanding the limits of algorithmic computation.