Provably computable functions and the fast growing hierarchy

Wainer, Stanley S.; Buchholz, Wilfried

doi:10.1090/conm/065/891248

Cited by 46 publications

(59 citation statements)

References 9 publications

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“…Hence, by a classical result about the provably recursive functions of P A (see, for example, [2], [5], [8], [17] for a proof), there is an α < ε 0 such that r * fε 0 (n + 1, n (n) , 3 n (n · 2 + 3)) ≤ H α (n) for n ≥ 0. Assertion (3) of Theorem 1 yields H ε0 (n − 2) ≤ r * fε 0 (n + 1, n (n) , 3 n (n · 2 + 3))…”

Section: By (2) and (3) Hence P(βmentioning

confidence: 92%

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A classification of rapidly growing Ramsey functions

Weiermann¹

2003

Proc. Amer. Math. Soc.

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Abstract. Let f be a number-theoretic function. A finite set X of natural numbers is called f -large if card(X) ≥ f (min(X)). Let P H f be the Paris Harrington statement where we replace the largeness condition by a corresponding f -largeness condition. We classify those functions f for which the statement P H f is independent of first order (Peano) arithmetic P A. If f is a fixed iteration of the binary length function, then P H f is independent. On the other hand P H log * is provable in P A. More precisely let fα(i) :where | i | h denotes the h-times iterated binary length of i and H −1 α denotes the inverse function of the α-th member Hα of the Hardy hierarchy. Then P H fα is independent of P A (for α ≤ ε 0 ) iff α = ε 0 .

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Section: By (2) and (3) Hence P(βmentioning

confidence: 92%

“…Without proof we mention that similar results also hold for the hydra battle and the Goodstein process. (See, for example, [11], [5] for an introduction into these topics.) Related results for Friedman style miniaturizations can be found in [18].…”

Section: Corollary 4 P H Log * Is Provable In P Amentioning

confidence: 99%

A classification of rapidly growing Ramsey functions

Weiermann¹

2003

Proc. Amer. Math. Soc.

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“…For a proof-theoretic approach (via iterated reflection principles) we refer to [19, Theorem 1, Proposition 7.3, Remark 7.4]. The author has found no fully explicit argument that the formalization is uniform in n. We provide a detailed proof of this fact in [20]: This is a proof-theoretic argument, formalizing the infinitary proof system from [21] by the method of [22].…”

Section: The Provably Total Functions Of Slow Reflectionmentioning

confidence: 99%

Proof lengths for instances of the Paris–Harrington principle

Freund

2017

Annals of Pure and Applied Logic

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As Paris and Harrington have famously shown, Peano Arithmetic does not prove that for all numbers k, m, n there is an N which satisfies the statement PH(k, m, n, N ): For any k-colouring of its n-element subsets the set {0,...,N − 1} has a large homogeneous subset of size ≥ m.A t t h e s a m e time very weak theories can establish the Σ 1 -statement ∃ N PH(k, m, n, N )f o r any fixed parameters k, m, n. Which theory, then, does it take to formalize natural proofs of these instances? It is known that ∀ m ∃ N PH(k, m, n, N )h a s a natural and short proof (relative to n and k)b yΣ n−1 -induction. In contrast, we show that there is an elementary function e such that any proof of ∃ N PH(e(n), n +1, n, N )b yΣ n−2 -induction is ridiculously long. In order to establish this result on proof lengths we give a computational analysis of slow provability, a notion introduced by Sy-David Friedman, Rathjen and Weiermann. We will see that slow uniform Σ 1 -reflection is related to a function that has a considerably lower growth rate than F ε0 but dominates all functions F α with α<ε 0 in the fast-growing hierarchy.

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“…Hence, we obtain a new proof of a well-known theorem due to Schwichtenberg and Wainer (cf. [34,25]). Consider now a more exotic theory IΣ 1 + IΠ − 2 .…”

Section: Smooth Turing Progressionsmentioning

confidence: 99%