The present work explores the theoretical limits of Machine Learning (ML) within the framework of Kolmogorov's theory of Algorithmic Probability, which clarifies the notion of entropy as Expected Kolmogorov Complexity and formalizes other fundamental concepts such as Occam's razor via Levin's Universal Distribution. As a fundamental application, we develop Maximum Entropy methods that allow us to derive the Erd\H{o}s--Kac Law and Hardy--Ramanujan theorem in Probabilistic Number Theory, and establish the impossibility of discovering a formula for primes using Machine Learning via the Prime Coding Theorem.

翻译：本文在科尔莫戈罗夫算法概率理论的框架内探讨了机器学习（ML）的理论极限，该理论将熵的概念阐释为期望科尔莫戈罗夫复杂度，并通过莱文通用分布形式化了奥卡姆剃刀等其他基本概念。作为一项基础应用，我们发展了最大熵方法，从而推导出概率数论中的埃尔德什—卡克定律与哈代—拉马努金定理，并通过素数编码定理确立了使用机器学习无法发现素数公式的结论。