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New Proofs Probe the Limits of Mathematical Truth


By proving a broader version of Hilbert’s famous 10th problem, two groups of mathematicians have expanded the realm of mathematical unknowability.

A few years later, Alan Turing and others built on his work, showing that mathematics is riddled with “undecidable” statements — problems that cannot be solved by any computer algorithm. Now, Koymans and his longtime collaborator, Carlo Pagano of Concordia University in Montreal — as well as another team of researchers working independently — have taken a major step toward that goal. Over the years, Shlapentokh and other mathematicians figured out what terms they had to add to the Diophantine equations for various kinds of rings, which allowed them to demonstrate that Hilbert’s problem was still undecidable in those settings.

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