However, the set of primitive recursive functions is not the largest recursively enumerable subset of the set of all total recursive functions. For example, the set of provably total functions (in Peano arithmetic) is also recursively enumerable, as one can enumerate all the proofs of the theory. While all primitive recursive functions are provably total, the converse is not true.
However the set of primitive recursive functions is not the largest recursively enumerable subset of the set of all total recursive functions For example the set of provably total functions (in Peano arithmetic) is also recursively enumerable as one can enumerate all the proofs of the theory While all primitive recursive functions are provably total the converse is not true