Why are not all recursively enumerable sets, recursive sets?

The complements of recursively enumerable but not recursive sets are all not recursively enumerable, although complements of sets that are not recursively enumerable are not necessarily recursively enumerable. read more

One set that is known to be recursively enumerable but not recursive is the set of consequences of the axioms of arithmetic (assuming these axioms are consistent, which no one really doubts although it's unprovable). read more

The set of "recursive languages" or "recursive sets" are sets where you can write a program that tells you whether the given input is in the set or not. All recursive languages are also recursively enumerable because you can just enumerate every string, and then output it if it's in your set. read more

The integers you get that correspond to syntactically correct Turing Machine encodings will be a recursive set of integers, say T (they can be checked by a TM "compiler"). But the subset H of T corresponding to TMs that halt when actually run will not be recursive, but it will be recursively enumerable. read more