Abstract

Let all finite sequences of letters of the English alphabet be ordered serially so that shorter sequences precede longer ones, and sequences of the same length are ordered alphabetically. Thus, an ordinal number is assigned to every possible English sentence of finite length. Among these numbers, some--call them S-numbers --will be the numbers of sentences which are instructions for writing down an infinite sequence of numbers. Now, it is impossible to state, in an English sentence of finite length, a generally applicable procedure for determining whether an arbitrarily given number is an S-number.