1. The Phenomenon of Statistical Stability and Its Properties

Here we examine the main manifestations of the phenomenon of statistical stability: the statistical stability of the relative frequency and sample average. Attention is drawn to an emergent property of the phenomenon of statistical stability. We discuss the hypothesis of perfect (absolute or ideal) statistical stability, which assumes the convergence of relative frequencies and averages. Examples of statistically unstable processes are presented. We discuss the terms “identical statistical conditions” and “unpredictable statistical conditions”. Hilbert’s sixth problem concerning the axiomatization of physics is then described. The universally recognized mathematical principles of axiomatization of probability theory and mechanics are considered. We propose a new approach for solution of the sixth problem, supplementing the mathematical axioms by physical adequacy hypotheses which establish a connection between the existing axiomatized mathematical theories and the real world. The basic concepts of probability theory and the theory of hyper-random phenomena are considered, and adequacy hypotheses are formalized for the two theories. Attention is drawn to the key point that the concept of probability has no physical interpretation in the real world.