Dynamic Hierarchies and the Limits of Reflection: A Metamathematical Model with Push and Pop Operations

The classical results of Gödel and Tarski show that no sufficiently strong formal theory can contain its own truth predicate. Feferman's transfinite progressions provide a way to define truth for each previous level on the next one, but the hierarchy only expands. This paper introduces dynamic hierarchies that allow not only expansion (push) but also contraction (pop), i.e., the deletion of the last added level. We prove that such hierarchies can be constructed for recursive ordinals while preserving consistency and recursive axiomatizability. The main results are: (i) semantic isolation after a pop - the resulting theory cannot define truth for the language it has just abandoned; (ii) limit collapse for oscillating sequences - under simple alternating rules, the limit theory reduces to the base system Q; (iii) a characterization of limit theories via the Fréchet filter. We interpret the collapsed limit as a state of formal silence: a theory stripped of all truth predicates, unable to express its own semantics. Connections to Feferman's hierarchies, reflection principles, and the philosophical notions of limits of introspection are discussed.