Streamlined subrecursive degree theory

Proof lengths for instances of the Paris–Harrington principle

“…Further results on slow provability can be found in work of Henk and Pakhomov [13]. To conclude this paper, let us rephrase our computational analysis in terms of subrecursive degree theory (see [24]): Proof. First we must verify that F ⋄ ε0 and F ε0 are honest functions (in the sense of [24]).…”

“…We need to see that this remains valid when one adds the base function F ⋄ ε0 (both as an axiom and as initial function of the fast-growing hierarchy). Indeed a general result to this effect is shown as part of the proof of [24,Theorem 16]. However, in [24] the approach to the fast-growing hierarchy is somewhat different: The paper works with norms of ordinals rather than explicit fundamental sequences.…”

“…We remark that the use of [24,Theorem 16] is, in some sense, a detour: Rather than considering provability in Peano Arithmetic one could use the "Generalized Growth Theorem" [24,Theorem 13] in combination with Proposition 3.8 above. Then, however, one has the technical task to reconcile the different definitions of the fast-growing hierarchy in our paper and in [24].…”

“…However, in [24] the approach to the fast-growing hierarchy is somewhat different: The paper works with norms of ordinals rather than explicit fundamental sequences. This appears to be a technicality, but rather than working out a detailed comparison we take the more direct way and reprove [24,Theorem 16] in our setting: As basis for our proof we take the analysis of the provably total functions of Peano Arithmetic in [21]. We assume that the reader has access to this paper.…”

“…To conclude this paper, let us rephrase our computational analysis in terms of subrecursive degree theory (see [24]): Proof. First we must verify that F ⋄ ε0 and F ε0 are honest functions (in the sense of [24]). We already know that the two functions are monotone and have elementary graphs (since they can be defined by ∆ 0 -formulas).…”