We show that the existence of a recursively enumerable set whose Turing degree is neither low nor complete cannot be proven from the basic axioms of first order arithmetic (P -) together with Σ 2 -collection (BΣ 2 ). In contrast, a high (hence, not low) incomplete recursively enumerable set can be assembled by a standard application of the infinite injury priority method. Similarly, for each n, the existence of an incomplete recursively enumerable set that is neither low n nor (...) high n - 1 , while true, cannot be established in P - + BΣ n + 1 . Consequently, no bounded fragment of first order arithmetic establishes the facts that the high n and low n jump hierarchies are proper on the recursively enumerable degrees. (shrink)