# Consistent, complete axiom system that proves its own consistency

Is there a consistent, complete axiom system that proves its own consistency?

I know that this question isn't exact and I haven't defined when an axiom system proves its own consistency because that's just human interpretation.

• The question was asked by Hilbert and answered in the negative by Gödel en.m.wikipedia.org/wiki/Gödel%27s_incompleteness_theorems – Jonathan Julián Huerta Jun 7 '15 at 16:28
• That wouldn't be a contradiction to a positive answer of my question. I read Gödel's original paper. – asdfusername Jun 7 '15 at 16:30
• @Jonathan: It would be a contradiction if the theory is also recursively enumerable. – Asaf Karagila Jun 7 '15 at 16:35
• No, that's wrong. – asdfusername Jun 7 '15 at 16:37
• You're right, it needs to also interpret arithmetic. But how do you plan on formulating "$T$ is consistent" without interpreting arithmetic? – Asaf Karagila Jun 7 '15 at 16:58