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Logic — Reverse mathematics cascade report

Cascade: A ∘ I ∘ C ∘ K ∘ N ∘ J ∘ M (seven classes) Partition: #23 of PR #677 — opens Logic section Status: verdict (a) SURVIVES — Big Five subsystems = 5-level hierarchy; Friedman's Grand Conjecture remains OPEN

Big Five subsystems (Class I cyclic-ordinal hierarchy)

Level Subsystem Theorems equivalent
1 RCA₀ Pigeonhole, IVT
2 WKL₀ Heine-Borel, Brouwer fixed-point, Hahn-Banach
3 ACA₀ Bolzano-Weierstrass, Ramsey RT²_k — Hurwitz triadic level
4 ATR₀ Open determinacy, perfect set theorem
5 Π¹₁-CA₀ Cantor-Bendixson on Polish spaces

Framework reading: the Big Five form a 5-level Class I cyclic-ordinal hierarchy. Level 3 ACA₀ sits at the Hurwitz triadic anchor — and Bolzano-Weierstrass (perhaps the most-used theorem of real analysis) sits at exactly that level. The framework predicts: ordinary mathematics concentrates AT the Hurwitz triadic level (level 3); higher levels (4, 5) are needed only for advanced descriptive set theory.

Friedman's Grand Conjecture (OPEN): all "concrete" ZFC mathematics is provable in EFA (Elementary Function Arithmetic, much weaker than even RCA₀). If true, working math IS substrate-perfect-math at very weak axiom strength — exactly the framework's substrate-DoF saturation reading.

Verdict

(a) SURVIVES — Big Five is a structurally clean Class I cyclic hierarchy; level-3 ACA₀ = Hurwitz triadic anchor matches Bolzano-Weierstrass. Friedman's conjecture remains OPEN.

Sources