Definitely his Incompleteness Theorem (I there are two of them and also a Completeness Theorem) was more general than what I mentioned but the shock was its app

Definitely his Incompleteness Theorem (I there are two of them and also a Completeness Theorem) was more general than what I mentioned but the shock was its application to arithmetic. The second order axioms are not recursively enumerable (think computable). It was the goal of Hilbert to find a computable system capable of proving (in theory) all true statements. There is no such computable system. One can find complete axiomatic systems of the integers. It's quite easy. Just take all true statements as the axiom system. The problem is that there is no computable method for determining in such a system whether or not a given statement is an axiom. Again, we humans can work in the second order axiom system but computers can't. So it appears there's a difference between human intelligence and computer intelligence.