Gödel's proven incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic.
The first incompleteness theorem states that NO CONSISTENT SYSTEM of axioms whose theorems can be listed by an "effective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of PROVING ALL TRUTHS about the relations of the natural numbers (arithmetic). For any such system, there will ALWAYS be statements about the natural numbers that are true, but that are UNPROVABLE within the system. The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency.