Gödel's incompleteness theorems are really something. Ultimately (iirc) it boils down to this:
If an axiomatic system is complex enough, it cannot be both consistent and complete.
I.e. either you have statements that you can't prove despite them being true, or your system is inconsistent (i.e. you can prove false statements).
That is a simple description of the first incompleteness theorem. The 2nd incompleteness theorem goes along similar lines:
If an axiomatic system is complex enough and consistent, it can't prove it's own consistency.
I.e.: If a system can prove it's own consistency, it isn't consistent.
Regarding how complex an axiomatic system has to be: Simple arithmetic is enough.
(FYI: it has been a while since I've spent time with the incompleteness theorems. No guarantees on accuracy, and correcting comments are welcome :) )
