Incompleteness Theorem for Computable Problems

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 \(\overline{K }=\left\{\left.x\right|x\notin {W}_{x}\right\}\) $\overline{K}=\left\{\left(x\right|,x,\notin ,W_x\right\}$ is not recursively enumerable. None of the computable functions can enumerate all the elements of the set \(\overline{K }\) $\overline{K}$ . As a result of the productivity of the set \(\overline{K }\) $\overline{K}$ , 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.