Namely, to establish the unsolvability of a decision problem, one has to reduce to it another problem the unsolvability of which has already been shown. In the first few chapters of this book, we reduce to Hilbert's Tenth Problem successively more and more complicated problems. This chain of reductions should lead ultimately to a problem for which we can give a direct proof of unsolvability. However, to be able to give such a proof, we need some way of surveying all conceivable methods of solving the problem. The possibility of doing this became available only after the development of the mathematically rigorous general notion of computability.
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