Abstract. In the theory of recursive functions, a recursively enumerable (but not recursive) set K = {x |x ∈ W_x } is obtained by Cantor’s diagonal method. Based on Diophantine sets, K is expressed by some polynomial that has positive roots. On the contrary, the set K  = {x |x ∉ W_x } is not recursively enumerable. None of the computable functions can enumerate all the elements of the set K . As a result of the productivity of the set K , a parameter exists for which the polynomial has no positive roots. However, it is impossible to prove their absence since this parameter does not belong to any recursively enumerable set.

Keywords: diagonal method, Diophantine set, recursively enumerable sets.