Chapter 4 Limits of Computability or Why

In Chapter 3 we discovered that there exist different infinite sizes. For instance, the number of real numbers is a larger infinity than the number of natural numbers. An infinite set is exactly as large as N if one can number the elements of A as the first one, the second one, the third one, etc. Here we aim to show that computing tasks exist that cannot be solved by any algorithm. The idea of our argument is simple. We show that the number of different tasks (computing problems) is a larger infinity than the number of all programs. Hence, there exist problems that cannot be algo