TITLE: Is consistency an axiom of mathematics?
QUESTION [25 upvotes]: I watched the numberphile video on Gödel's Incompleteness Theorem today, and I was wondering about something. 
It seems the key to accepting the truth of Gödel's Theorem is to demand that mathematics is consistent. However, isn't this like invoking consistency as an axiom of mathematics? Therefore, aren't we proving the truth of Gödel's Theorem using the axioms? Is the consistency of mathematics something other than an axiom?
I've read elsewhere on here that another way to state Gödel's theorem is to say that no formal mathematics system can prove it's own consistency. Does that mean we just have to assume our system of mathematics is consistent?
Definitely not an expert in this!

REPLY [3 votes]: I would not say that "the key to accepting the truth of Gödel's Theorem is to demand that mathematics is consistent." The theorem says that, if a formal system is consistent, then it can be used to express true statements that don't have proofs within the system. So, even if mathematics were inconsistent, the theorem would still be true.