alt Hacker NewsYour very first sentence is simply wrong, I don't know what more there is to say other than that you clearly don't understand what Gödel's results are about (hint: they're called "incompleteness theorems", not "contradiction theorems"). Maybe read a textbook? I could recommend several good ones.
The other sentences basically fall in the category of "not even wrong", i.e. basically nonsensical.
Godel starts by the assumption that the system is free from contradictions which makes his papers largely irrelevant here.
Again, feel free to explain how you can have a second order logical system without axiomatic limitations that doesn't contain contradictions.
Hell, go by the more popular version of first order logic + set theory if you're more comfortable with that.