Do we know that the P vs. NP question isn't affected by Gödels incompleteness theorem?
Note that G&ouml;del's Incompleteness Theorems are the following statements:


Any formal system which can be used to express arithmetic is either incomplete (there are statements which are neither provable nor disprovable) or inconsistent.
Any formal system which can be used to prove a statement equivalent to  its own consistency, is in fact inconsistent.


But assuming, for instance, that ZF set theory (or an essentially equivalent  foundation for mathematical subjects such as computer science) is consistent, we don't need G&ouml;del's Theorems anymore to tell us that there are statements which are neither provable nor disprovable from our foundations: interesting specific examples, such as the Axiom of Choice have already been demonstrated. So even without G&ouml;del's Theorem, it is certainly concievable that a statement such as "P = NP" is indepdendent of set theory.

For this question, Scott Aaronson has written a very accessible article on the subject, which also goes over the basics of formal logic, what it could mean to have a model of set theory which included an axiom which is equivalent to asserting its own inconsistency, and the subject of logical independence results in general.