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A team of researchers has stumbled on a question that is mathematically unanswerable because it is linked to logical paradoxes discovered by Austrian mathematician Kurt Gödel in the 1930s that can’t be solved using standard mathematics.

The mathematicians, who were working on a machine-learning problem, show that the question of ‘learnability’ — whether an algorithm can extract a pattern from limited data — is linked to a paradox known as the continuum hypothesis. Gödel showed that the statement cannot be proved either true or false using standard mathematical language. The latest result appeared on 7 January in *Nature Machine Intelligence*^{1}.

“For us, it was a surprise,” says Amir Yehudayoff at the Technion–Israel Institute of Technology in Haifa, who is a co-author on the paper. He says that although there are a number of technical maths questions that are known to be similarly ‘undecidable’, he did not expect this phenomenon to show up in a relatively simple problem in machine learning.

John Tucker, a computer scientist at Swansea University, UK, says that the paper is “a heavyweight result on the limits of our knowledge”, with foundational implications for both mathematics and machine learning.