Challenges for the periodic systems of elements: chemical, historical and mathematical perspectives

Unveiling numerical trends among either atomic or equivalent weights that somehow preserved resemblances among elements was frequent in the 1860s. Standing out from the crowd, Meyer and Mendeleev went beyond numerical relationships, certainly motivated by a pedagogical aim. Both were after a system synthesizing the chemical knowledge of their times in an appealing way to be presented to chemistry students. Is it still the periodic system aiming at that? Is it really a map of the current chemistry landscape? Solving these questions entails addressing others such as: what is the periodic system? If it is about chemical elements, do we really know what a chemical element is? Is the system unique? How was and how is currently built up? Is it actually used to conduct chemical research? A suitable tool for chemical predictions? Or is it just a mnemotechnic for fancy trends? Does it have a limit, or multiple ones, instead? Let us try to address these questions and let us begin by analysing the concept of chemical element.