Toward a Theory of Hierarchical Definability (THD)
Attempts to formulate mathematically precise definitions of basic concepts such as causality, randomness and probability have a long history. The concept of hierarchical definability that is outlined in the following suggests that such definitions may not exist. Furthermore, it suggests that existing definitions of many basic concepts, among them those of linearity stability, statistical independence and Pareto-optimality, may be in need of reformulation.In essence, definability is concerned with whether and how a concept, X, can be defined in a way that lends itself to mathematical analysis and computation. In mathematics, though not in logic, definability of mathematical concepts is taken for granted. But as we move further into the age of machine intelligence and automated reasoning, the issue of definability is certain to grow in importance and visibility, raising basic questions that are not easy to resolve.