The Halting Problem

Theorem 3:

Let M₁, M₂, ... be a list of all TM using 1, #.

The Halting Problem is the problem of finding an effective procedure to compute function h such that h(m,n) = 1 iff M_m doesn't halt if it begins by scanning n 1s.

The Halting Problem is unsolvable.

Proof:

If it were solvable then the SHP would be solvable with S(n) = h(n, n).