Non-Well-Founded Hierarchies of Nested Systems

We develop a set-theoretic framework for systems organised across scales of description. A system is a set, its carrier, partitioned into elements and a passive medium, together with valued relations among its elements. Dependence between systems is formalised as _generation_ over a set of admissible configurations: one system is generated by another when its relational data factor through the other’s. We show that generation is characterised by inclusion of equivalence kernels and that, evaluated at a single configuration, it is vacuous; functional dependence between levels is a property of variables, not of values. On this basis we define subsystems, enclosure (parthood together with membership), superlevels as aggregations whose data are generated from the level below, and hierarchies as level-graded families of systems with partial level transitions. We prove that level transitions are strict, that enclosure is anti-transitive, and that an up-then-down round trip recovers a system up to inter-derivability if and only if its aggregation is faithful on the admissible configurations. Non-well-founded descent, by enclosure or by refinement of levels, is consistent with ordinary set theory but forces infinite carriers; for finite carriers we give sharp bounds on the length of descent. A dyadic hierarchy shows that bi-infinite hierarchies exist.