An extremely sharp phase transition threshold for the slow growing hierarchy

We investigate natural systems of fundamental sequences for ordinals below the Howard Bachmann ordinal and study growth rates of the resulting slow growing hierarchies. We consider a specific assignment of fundamental sequences which depends on a non negative real number ε. We show that the resulting slow growing hierarchy is eventually dominated by a fixed elementary recursive function if ε is equal to zero. We show further that the resulting slow growing hierarchy exhausts the provably recursive functions of ID1 if ε is strictly greater than zero. Finally we show that the resulting fast growing hierarchies exhaust the provably recursive functions of ID1 for all non negative values of ε. Our result is somewhat surprising since usually the slow growing hierarchy along the Howard Bachmann ordinal exhausts precisely the provably recursive functions of PA. Note that the elementary functions are a very small subclass of the provably recursive functions of PA and the provably recursive functions of PA are a very small subclass of the provably recursive functions of ID1. Thus the jump from ε equal to zero to ε greater than zero is one of the biggest jumps in growth rates for subrecursive hierarchies one might think of. This article is part of our general research program on phase transitions in logic and combinatorics. Phase transition phenomena are ubiquitous in a wide variety of branches of mathematics and neighbouring sciences, in particular, physics (see, for example, [6]). An informal description of a ‘phase transition effect’ is the effect behaviour wherein ‘small’ changes in certain parameters of a system occasion dramatic shifts in some globally observed behaviour of the system, such shifts being marked by a ‘sharp threshold point’. An everyday life example of this is the change from one material state to a different one as temperature is increased, with the ‘threshold’ being given by melting/boiling point. Similar phenomena occur in mathematical and computational contexts like evolutionary ? The author is a Heisenberg fellow of the DFG and has been supported in part by DFG grant We 2178 6/1 graph theory (see, e.g., [3, 10]), percolation theory (see, e.g., [9]), computational complexity theory and artificial intelligence (see, for example, [7, 11]). The purpose of PTLC is to study Phase Transitions in Logic and Combinatorics. We are particularly interested in the transition from provability to unprovability of a given assertion by varying a threshold parameter. On the side of hierarchies of recursive functions this reduces to classifing the phase transition for the growth rates of the functions involved. In this article we are concerned with phase transitions for the slow growing hierarchy and we continue the investigations from [12–15]. From the pure logical side this article is motivated by the classical classification problem for the recursive functions and the resulting problem of comparing the slow and fast growing hierarchies. It has been claimed, for example in [3] p. 439 l.-5, that for sufficiently big prooftheoretic ordinals the slow and fast growing hierarchies will match up. The results of this paper may indicate that this claim might not be true in general. To formulate the results precisely we introduce some notation. For an ordinal α less than the Howard Bachmann ordinal let Nα be the number of symbols in α which are different from 0 and +. The idea is essentially that Nα is the number of edges in the tree which represents the term for α. For a limit ordinal λ let λ[x] := max{β < λ : Nβ ≤ Nλ + x}. This assignment of fundamental sequences is natural and does not change, as we will show in the appendix, the growth rate of the induced fast growing hierarchy. But, as our first main theorem shows, the induced slow growing hierarchy (along the Howard Bachmann ordinal) consists of elementary functions only. This generalizes results from [4] where we showed that the resulting slow growing hierarchy along Γ0 consists of elementary recursive functions only. At first sight the resulting slow growing hierarchies seem always to collapse under this assignment of fundamental sequences and one may wonder how robust this phenomenon is. We prove therefore in a separate section a very surprising and extremely sharp phase transition threshold for the slow growing hierarchy. The upshot is that small changes prevent the hierarchies from collapsing. For a given real number ε ≥ 0 let λ[x]ε := max{β < λ : Nβ ≤ (1 + ε) · Nλ + x}. Then, as we just said, for ε = 0 the resulting slow growing hierarchy is very slow growing but for any ε > 0 the resulting slow growing hierarchy becomes fast growing and matches up with the fast growing hierarchy at all -numbers below the HowardBachmann ordinal. We conjecture that within the phase transition, i.e. when in the definition of λ[x]ε the number ε is a function of λ and x, we may arrange other behaviours of the resulting slow growing hierarchy. The paper is not fully self contained. The proof of the first main theorem requires basic familiarity with Buchholz style notation systems for the Howard Bachmann ordinal. (Knowledge of [4] is more then sufficient.) The proof of the second main result should be generally accessible (at least when one restricts the consideration to ordinals below ε0. 1 Proof of the first main result