Slow Growing Versus Fast Growing

I falsely claimed, as an aside remark in [8] and also implicitly in the abstract [9], that the slow-growing hierarchy “catches up” with the fast-growing hierarchy at level Γ 0 , i.e. that, for all x > 0, where x ′ is some simple (even linear) function of x . Girard [4] gave the first correct analysis of the deep relationship which exists between G and F , based on his extensive category-theoretic framework for -logic. This analysis indicates that the first point at which G catches up with F is the ordinal of the theory ID ω (0 of arbitrary finite iterations of an inductive definition. This is very far beyond Γ 0 ! In particular, in order to capture F at level ∣ID n ∣ the slow-growing hierarchy must be generated up to ∣ID n +1 ∣, i.e. one extra iteration of an inductive definition is needed in order to generate sufficient new ordinals.