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Let me give a toy example of the relativization barrier. The canonical example is the time hierarchy theorem that ${\bf TIME}[t(n)] \subsetneq {\bf TIME}[t(n)^2]$. The proof (by diagonalization) is only a little more involved than the proof that the halting problem is undecidable: we define an algorithm $A(x)$ which simulates the $x$th algorithm $A_x$ on ...


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How about just a blanket statement with no assumptions: A and B may be the same part of C This would seem to be valid by the same reasoning! However it is only so if "may" ranges over all models of the language. If, as is usual, "may" only ranges over the possible worlds in a single model, it is not valid.


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