Lev Reyzin in Nature:
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. Writing in Nature Machine Intelligence, Ben-David et al.5 show that the field of machine learning, although seemingly distant from mathematical logic, shares this limitation. 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. The power of this idea is illustrated by the following example: although it seems hopelessly difficult to explicitly program a computer to determine what objects are in a picture, the Viola–Jones machine-learning system can detect human faces in real time after being trained on a labelled sample of photographs6. Today, we regularly interact with machine-learning algorithms, from virtual assistants on our phones to spam filters for our e-mail. But these modern real-world applications trace their origins to a subfield of machine learning that is concerned with the careful formalization and mathematical analysis of various machine-learning settings.