|Godel’s Second Incompleteness Theorem. In any consistent axiomatizable theory (axiomatizable means the axioms can be computably generated) which can encode sequences of numbers (and thus the syntactic notions of “formula”, “sentence”, “proof”) the consistency of the system is not provable in the system.|
|The theories of real numbers, of complex numbers, and of Euclidean geometry do have complete axiomatizations. Hence these theories have no true but unprovable sentences. The reason they escape the conclusion of the first incompleteness theorem is their inadequacy, they can’t encode and computably deal with finite sequences.|
|There is a weak theological parallel in the Problem of Evil:
Actually this doesn’t prove divine nonexistence: just that certain notions of being “ultimately responsible” and being “perfectly just” are inconsistent. Being ultimately responsible is a form of strength like being able to encode sequences of numbers. Being just is like the property of being self-consistent (inconsistency is the sole mathematical evil). As with diagonal arguments, you can’t have both.