 The existence of incomplete theories is hardly surprising. Take any theory, even a complete one (see below for examples), and drop some axiom; unless the axiom is redundant, the resulting system is incomplete. The incompleteness theorems, however, deal with a much more radical kind of incompleteness phenomenon. Unlike the above sort of trivially incomplete theories, which can be easily completed, there is no way of completing the relevant theories; all their extensions, inasmuch as they are still formal systems and hence axiomatizable, are also incomplete. They remain, so to speak, eternally incomplete and can never be completed. They are “essentially incomplete”. 