I thought of trivial proof of something related to Godel's second incompleteness theorem. According to Wikipedia "The second incompleteness theorem, an extension of the first, shows that such a system [i.e. a system of axioms whose theorems can be listed by an effective procedure] cannot demonstrate its own consistency." (Yes, I'm afraid that is my main source of knowledge about Godel's theorems). My thought is very simple. Suppose your system of axioms is inconsistent. Then you can construct a contradiction and use it to prove (by contradiction) any statement you choose, including the statement that "this system of axioms is consistent". So even if you could demonstrate consistency from within the system, you could never trust the proof.