  Turing's proof can be recast in many ways, but the core idea depends on the self-reference involved in a machine operating on symbols, which is itself described by symbols and so can operate on its own description. Indeed, the self-referential aspect of the theory can be highlighted by a different form of the proof, which Turing preferred (Turing 1936, p. 247). Suppose that such a machine for deciding satisfactoriness does exist; then apply it to its own description number. A contradiction can readily be obtained. However, the ‘diagonal’ method has the advantage of bringing out the following: that a real number may be defined unambiguously, yet be uncomputable. It is a non-trivial discovery that whereas some infinite decimals (e.g. π) may be encapsulated in a finite table, other infinite decimals (in fact, almost all) cannot. Likewise there are decision problems such as ‘is this number prime?’ in which infinitely many answers are wrapped up in a finite recipe, while there are others (again, almost all) which are not, and must be regarded as requiring infinitely many different methods. ‘Is this a provable proposition?’ belongs to the latter category.