During the twentieth century, discoveries in mathematical logic revolutionized our understanding of the very foundations of mathematics. In 1931, the logician Kurt Gödel showed that, in any system of axioms that is expressive enough to model arithmetic, some true statements will be unprovable1. And in the following decades, it was demonstrated that the continuum hypothesis -- which states that no set of distinct objects has a size larger than that of the integers but smaller than that of the real numbers -- can be neither proved nor refuted using the standard axioms of mathematics2–4. They identify a machine-learning problem whose fate depends on the continuum hypothesis, leaving its resolution forever beyond reach. Machine learning is concerned with the design and analysis of algorithms that can learn and improve their performance as they are exposed to data.