Admissible principles for the extensions Fn that preserve constructivity

Identify which principles—such as bar recursion, restricted choice, and definitional reflection—can be added to the conservative extensions Fn in the fractal countability hierarchy over a base system F0 while preserving constructive interpretability and avoiding non-constructive comprehension.

Background

Each stage Fn in the fractal countability framework is intended to be a constructive conservative extension of F0 that expands definability without collapsing into impredicativity. Candidate additions include bar recursion, restricted forms of choice, and definitional reflection, all of which can affect the resulting family Sn of definable subsets of N.

Determining admissibility criteria is crucial both to ensure that S∞ remains a constructively interpretable closure and to calibrate how far beyond HYP the hierarchy can extend while maintaining constructive fidelity.