A set is recursively enumerable if there exists a Turing machine that will enumerate all the elements of the set, possibly without halting if the set is infinite. This means that for any given element in the set, there is a systematic procedure to confirm its membership, but there may not be a way to decide membership for every possible input in a finite amount of time. The concept connects deeply with computational theory, particularly regarding which sets can be effectively calculated or listed by algorithms.
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