In such a case it may be possible to come across the program p by accident, but then we would still need to check that it was the correct program. Below we show that, given plausible limitations on our programming ability , such cases do exist. Thus the normal characterization of computability is too weak for the purposes of the arguments in this paper. If this is not to be the result of an unverifiable accident we must be able to somehow effectively construct the TM that computes L . We can formalise the situation in the following way.
L is a statement in some recursively axiomatized logic.
So the statements in this logic can be effectively enumerated . Similarly enumerate the possible TMs .