  Turing machines, like computer programs, are countable; indeed they can be ordered in a complete list by a kind of alphabetical ordering of their ‘tables of behaviour’. Turing did this by encoding the tables into ‘description numbers’ which can then be ordered in magnitude. Amongst this list, a subset of them (those with ‘satisfactory’ description numbers) are the machines which have the effect of printing out infinite decimals. It is readily shown, using a ‘diagonal’ argument first used by Cantor and familiar from the discoveries of Russell and Gödel, that there can be no Turing machine with the property of deciding whether a description number is satisfactory or not. The argument can be presented as follows. Suppose that such a Turing machine exists. Then it is possible to construct a new Turing machine which works out in turn the Nth digit from the Nth machine possessing a satisfactory description number. This new machine then prints an Nth digit differing from that digit. As the machine proceeds, it prints out an infinite decimal, and therefore has a ‘satisfactory’ description number. Yet this number must by construction differ from the outputs of every Turing machine with a satisfactory description number. This is a contradiction, so the hypothesis must be false (Turing 1936, p. 246). From this, Turing was able to answer Hilbert's Entscheidungsproblem in the negative: there can be no such general method.