Perhaps that is too strong a statement. How about "Godel has no
implications about the limits of our knowledge of reality."? I
reject your postulation that "The Goldbach Conjecture is either
true or false", except within the definitions of "true" and "false"
postulated by the system in which it is expressed. There is no
fundamental connection between the mathematical, human-made concept
of "true theorem" (i.e., theorem logically consistent with axioms)
and the metaphysical "truth" as in "consistent with reality". Any
such mapping we make between the two is an inductively-reasoned
convenience for making predictions about reality.
The final arbiter of reality is observation. When a mathematical
model (like, say, Newtonian mechanics) maps well enough to our
observations to make useful predictions, then we should use it. And
when it fails, we should modify it. Whatever "truths" we may find
by exploring mathematics itself, apart from reality, can have no
bearing upon our lives or experiences until we choose to make such
a mapping, and when we do, it is observation, not logical proof, that
will be our judge. Any "truth" that cannot be tested by perception
is of no possible value to any real being. Therefore, Godel's
result is simply a fact about the way we manipulate symbols. It has
no bearing whatsoever upon reality or our ability to perceive it.