General framework for comparing definability hierarchies from distinct base systems

Develop a general theory for comparing definability hierarchies induced by different base systems F0 and sequences of conservative extensions {Fn}, providing notions of translation, embedding, equivalence, or separation between the resulting closures S∞ to relate their definitional power across foundations.

Background

Because fractal countability is explicitly relative to a base system F0, different choices of F0 (e.g., RCA0 versus type-theoretic bases) may yield non-equivalent and potentially incomparable hierarchies. Establishing systematic methods to compare such hierarchies is essential for foundational pluralism and cross-system analysis.

A comparison framework would clarify when one hierarchy simulates or embeds into another, when they are equivalent on certain classes, or when they are provably incomparable, thereby mapping the landscape of stratified definability across diverse formal foundations.