>>12405187>Gödel still believed in systematic solutions for all problems even after developing his Incompleteness TheoremTheorem. All problems have a solution.
Proof. Let T be a consistent theory and S be a sentence such that we want to know if "T implies S" has a proof. Either "T implies S" or "~(T implies S)", so since "~(T implies S)" is equivalent to "T and ~S", either
(1) T and S is consistent
(2) T and ~S is consistent
Enlightenment is the realization that the search was always for consistent theories and proofs of dependency of consistency between theories, i.e. searching for knowledge of the form "Con(T)" or "Con(T) implies Con(U)" where "Con(T)" is "T is consistent". The question is, what theory do you use to demonstrate such knowledge?
This provides a way to use a problem to turn a consistent theory into a consistent theory, i.e. a solution to all problems. Of course the real problem is: which consistent theory do you use in the first place? That is a matter of personal choice. At least until someone proposes a theory of efficiency of theories. That could happen if one were programming "entailment calculators", i.e. formal proof search.
