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Informally speaking, Kurt Godel's Incompleteness Theorem for formal theories of arithmetic is based on the peculiar nature of self-referential sentences. Godel's Theorem showed that the massive project undertaken by Russell and Whitehead in Principia Mathematica to reduce all mathematical truths to logic was impossible.

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It's also the basic reason that no general solution to the halting problem can exist in computer science. Probably there is something analogous to be said about physics, which is by its nature self-referential.