User blog:Triakula/Uncomputable functions defined as a sequence of computable functions

Hello.

I want to come up with an explicit diagonalizing sequence of computable functions which does not use ordinals. Also, I present an uncomputable function which is defined from it. I appreciate to point me if it is well-defined.

Let \(f_i(n)\) to be \(i\)-th partial recursive function defined by some Turing machine using some fixed ordering of them.

Let's order functions by growth rate. Generally, it is not true that if \(i > j\), then \(f_i > f_j\).

I define the function \(g_i(n)\) by the following way: \(g_0(n) = f_0(n)\) and \(g_{n+1}(n)\) to be the function with least \(i\) so that \(f_i >^* g_n \land g_{n+1} >^* g_n\). For instance, if \(g_0(n) = f_0(n) = n \cdot 2\), \(f_k(n) = n^2\), and there is no function in form \(f_i(n)\) with \(i \in \{0,1,\cdots,k\}\) that eventually dominates \(n \cdot 2\), then \(g_1(n) = n^2\).

Then \(g_i(n)\) is a computable function by definition as it is \(f_j(n)\) for some Turing machine. The sequence \(g_0(n),g_1(n),g_2(n),\cdots\) forms a functional hierarchy for which each member is a computable function, but \(h(n) = g_n(n)\) is uncomputable function as it cannot be expessed by \(f_j(n)\) for fixed \(j\).