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Theory of computation tells us that there are some languages that cannot be recognized by a Turing machine. That is, there are well-defined problems for which no Turing machines can provide an algorithm which solves the problem.
Many problems that are undecidable seem to have a reduction to the Halting problem: Matyasevich's proof of the undecidability of Diophantine equations uses the Halting problem, the machine equivalence problem uses the Halting problem as well...
The question is: are all undecidable problems (in the sense of Turing) reducible to the Halting problem?
If so, is there a proof of this? If not, is there a counterexample of such a problem that provably cannot be reduced or mapped to the Halting problem.