Godel's Incompleteness Theorem is not quite the bludgeon some people (like Roger Penrose) think it is. It relies on certain assumptions that are not universally true in practice.
Would you agree that science must be self-consistent?It must not contradict itself? You may say that logical self-consistency has no impact on science..are you sure? I think many scholars would claim that not only have we admitted that there is an abstract supernatural realm within mathematics, but these results are not confined to mathematics. Logical self-consistency is crucial to science. The supernatural can have a testable basis and can be scientific. John Wilson, a anti-creationists, states "it is not surprising that unnatural and supernatural phenomena are not represented in the realm of current science, but that fact does not eliminate them from the PROVINCE of science.
note your use of the word "theoretical" Godel's Theorem is a bludgeon and one who respects honest discussion should admit it. Your examples need to be more specific and not be sending me or others down those illusions of facts .