Turing used the halting problem to demonstrate that there are some functions (or numbers) that one can describe but not compute. To see this, consider that, given any programming language, one can assign a unique integer to each program. E.g. by sorting the programs in order of length, and numbering them from 1 upwards. Now, imagine the real number whose th digit it 1 if the th program halts and otherwise. Since the halting problem can't be solved, it's impossible to compute this number. Nevertheless, it is true in fact that every program does either halt or fail to halt. Hence every digit of must be either 0 or 1. But we can never know which.