Gödel theorems statments:
1.In any consistent axiomatic system (formal system of mathematics) sufficiently strong to allow one to do basic arithmetic, one can construct a statement about natural numbers that can be neither proved nor disproved within that system
2.Any sufficiently strong consistent system cannot prove its own consistency.
Gödel's theorems are theorems in first-order logic, and must ultimately be understood in that context.
The existence of an incomplete system is in itself not particularly surprising. For example, if you take Euclidean geometry and you drop the parallel postulate, you get an incomplete system. An incomplete system can mean simply that you haven't discovered all the necessary axioms.
What I mean is that if you are starting from incomplete information, you are surely going to arrive to incomplete results. Gedel bases his calculations in first order propositional logic which is certainly not enough to prove that a simple math system is complete.
3 days ago