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：本文在柯尔莫哥洛夫算法概率理论框架下探讨了机器学习（ML）的理论极限。该框架通过列文通用分布将熵的概念明晰为期望柯尔莫哥洛夫复杂度，并形式化奥卡姆剃刀等基本概念。作为基础性应用，我们发展了最大熵方法，据此可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证毕利用机器学习发现素数公式的不可行性。