[kasperbn]

It really depends on what is meant by "objectively true or false".

You're right that some statements within a formal system are true or false.

However, the theorem does imply that the behavior of the "world out there" (and any explanation of it) is either contradictory or not following a fixed set of rules.

So yes, we can state something to be "objectively" true or false but only if we're willing to use a contradictory explanation - which we usually are in the name of usefulness, not in the name of "absolute objective truth".