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Investigations on slow versus fast growing: How to majorize slow growing functions nontrivially by fast growing ones

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Let T(Ω) be the ordinal notation system from Buchholz-Schütte (1988). [The order type of the countable segmentT(Ω)0 is — by Rathjen (1988) — the proof-theoretic ordinal the proof-theoretic ordinal ofACA 0 + (Π l1 TR).] In particular let ↦Ω a denote the enumeration function of the infinite cardinals and leta ↦ ψ0 a denote the partial collapsing operation on T(Ω) which maps ordinals of T(Ω) into the countable segment TΩ 0 of T(Ω). Assume that the (fast growing) extended Grzegorczyk hierarchy\((F_a )_{a \in T(\Omega )_0 }\) and the slow growing hierarchy\((G_a )_{a \in T(\Omega )_0 }\) are defined with respect to the natural system of distinguished fundamental sequences of Buchholz and Schütte (1988) in the following way:

$$\begin{array}{*{20}c} {G_0 (n): = 0,} & {F_0 (n): = (n + 1)^2 ,} \\ {\begin{array}{*{20}c} {G_{a + 1} (n): = G_a (n) + 1,} \\ {G_l (n): = G_{l[n]} (n),} \\ \end{array} } & {\begin{array}{*{20}c} {F_{a + 1} (n): = \underbrace {F_a (...F_a }_{n + 1 - times}(n)...),} \\ {F_l (n): = F_{l[n]} (n),} \\ \end{array} } \\ \end{array}$$

wherel is a countable limit ordinal (term) and (l[n]) n N is the distinguished fundamental sequence assigned tol. For each ordinal (term)a in T(Ω) and each natural numbern letC n (a) be the formal term which results from the ordinal terma by successively replacing every occurence ofψ a by\(\psi _{ - 1 + C_n (a)}\) whereψ −1 is considered as a defined function symbol, namely\(\psi _{ - 1} b: = F_{\psi _0 b + 1} (n + 1)\). (Note thatψ a 0=Ω a ) In this article it is shown that for each ordinal termψ 0 a in T(Ω) there exists a natural numbern 0 such thatψ 0 C n (a) ∈ T(Ω) and\(G_{\psi _0 a} (n) \leqslant F_{\psi _0 C_n (a) + 1} (n + 1)\) holds for alln≥n 0. This hierarchy comparison theorem yields a plethora of new results on nontrivial lower bounds for the slow growing ordinals — i.e. ordinals for which the slow growing hierarchy yields a classification of the provably total functions of the theory in question — of various theories of iterated inductive definitions (and subsystems ofKPi) and on the number and size of the subrecursively inaccessible ordinals — i.e. ordinals at which the extended Grzegorczyk hierarchy and the slow growing hierarchy catch up — below the proof-theoretic ordinal ofACA 0+(Π l1 TR). In particular these subrecursively inaccessibles ordinals are necessarily of the form\(\psi _0 \Omega ..._{\Omega _\omega }\).

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Weiermann, A. Investigations on slow versus fast growing: How to majorize slow growing functions nontrivially by fast growing ones. Arch Math Logic 34, 313–330 (1995). https://doi.org/10.1007/BF01387511

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