Dynamic Axiomatic Hierarchies: Push and Pop Operations in the Context of Self-Reference

The classical results of Gödel and Tarski demonstrate fundamental limitations of formal systems: no consistent recursively axiomatizable theory extending Robinson arithmetic Q can contain its own truth predicate. Feferman's transfinite progressions provide a hierarchical approach where truth for each level becomes definable at the next, yet these hierarchies only expand monotonically. This paper introduces a novel generalization - dynamic hierarchies - which incorporate not only expansion (push operations) but also contraction (pop operations) through deletion of previously added levels. We construct such hierarchies indexed by recursive ordinals and prove that consistency and recursive axiomatizability are preserved at every stage. The main result establishes semantic isolation following a pop: the resulting theory cannot express truth for the deleted level, formalized via a diagonal argument using only truth in the standard model N, independent of provability relations between theories. Limit theories are defined using the Fréchet filter, with their properties characterized based on the asymptotic behavior of pushes and pops. We demonstrate that in the absence of pops, the construction reduces exactly to Feferman's classical progressions. Several natural transition rules are examined, including alternating push/pop patterns and bounded stack depth, yielding limit theories that typically collapse back to Q. The framework provides a rigorous foundation for modeling reflexive processes with revision of principles, opening directions for investigating proof-theoretic ordinals, non‑monotonic hierarchies, and connections with axiomatic truth theories and reflection principles.