Precise comparison of the fractally countable closure S∞ with HYP, Delta^1_1, and the analytic hierarchy

Determine the precise relationships between the fractally countable closure S∞—defined as S∞ = ⋃_{n∈N} Sn where each Sn is the family of subsets of N definable in the conservative extension Fn of a base formal system F0—and the classical hierarchies HYP (the hyperarithmetical sets), Delta^1_1, and the analytic hierarchy; in particular, ascertain whether there exist base systems F0 and sequences of conservative extensions {Fn} for which S∞ is strictly stronger or strictly weaker in definitional power than HYP or Delta^1_1, or occupies a distinct position relative to levels of the analytic hierarchy.

Background

The paper introduces fractal countability as a layered, process-relative notion of definability. Given a base system F0 and a sequence of conservative extensions {Fn}, each stage Sn consists of subsets of N definable in Fn, and the closure S∞ = ⋃n Sn remains countable but can expand in expressive power as n increases.

Section 3.2 shows that with a suitable choice of {Fn} one can reconstruct HYP, but the framework is not intrinsically tied to oracle computations and can potentially extend beyond HYP depending on which constructive principles are admitted at successive stages. This raises the need to situate S∞ precisely relative to established hierarchies such as HYP, Delta^1_1, and the broader analytic hierarchy, and to identify situations in which S∞ is strictly stronger or weaker than these benchmarks.