[kasperbn]

Which is quite significant I would say. Suddenly we can no longer talk about what is objectively true and false, only what is subjectively useful. That last sentence is of course also neither objectively true nor false, just like this one. See it's a mess :)

[teach]

Gödel's incompleteness theorem does not imply that NOTHING can be objectively true or false.

[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".