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Decision problems for which it is unknown whether they are decidable
In computability theory, what are examples of decision problems of which it is not known whether they are decidable?
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Any important consequences with presupposition of $\mathbf{P} \neq \mathbf{NP}$
As we know, there are lots of consequences with the presupposition of the Riemann Hypothesis.
Similarly, are there any important consequences with the presupposition of $\mathbf{P} \neq \mathbf{NP}$ ?...
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