Say we had a Turing machine M with states q1, ... qR, with a tape alphabet with symbols s1, ... sm, with the blank denoted by s0, and transitions giving the current state, current symbol, and actions performed (which might be to overwrite the current tape symbol and move the tape head left or right, or maybe not move it at all), and the next state.
The fact that such a number always exists for any Turing machine is generally the important thing.
By making use of a technique similar to Cantor's diagonal argument, it is possible exhibit such an uncomputable function, for example, that the halting problem in particular is undecidable.
First, let us denote by U(e, x) the action of the universal Turing machine given a description number e and input x, returning 0 if e is not the description number of a valid Turing machine.
From this definition δ is defined for every input and must naturally be total recursive.