In 1931 a 25-year-old Austrian mathematician, Kurt Gödel, developed the ‘Incompleteness Theorem’ as he analysed mathematical formulae and their relevancy. The theory is still valid today and has never been refuted, based as it is on simple common sense and demonstrable logic which parametrises the limits of provability in formal axiomatic and mathematical theories.
Principle 1: in a consistent and formal system named F, within which mathematical operations are carried out, there are statements of the language within F, which can be neither proven nor disproven (unless one uses an outside framework reference, or context, see the simple examples given above) and,
Principle 2: in this formal system named F, we cannot prove that the system itself is consistent, we must assume that it is intern
ally consistent. The internal consistency of the system has a great impact on the mathematical operations but cannot be verified without the use of an external frame of reference.