Recursively enumerable sets

Recursively enumerable (r.e.) sets are collections of items that a computer program can list or generate over time, but may never finish. If an element belongs to the set, the program will eventually find and output it; if it doesn’t belong, the program runs indefinitely without confirming exclusion. Think of it like a search process that gradually finds valid items—if something is in the set, it will be revealed eventually; if not, the search just continues forever. These sets are fundamental in computability theory, describing what can be effectively enumerated or recognized by algorithms.