An extremely sharp phase transition threshold for the slow growing hierarchy

Abstract

Abstract. 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 P A. 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.

Unprovability results involving braids

“…x where f is a very slowly increasing function. Analogous to the results of [32,33,34], Theorem B is a typical example of a so-called phase transition phenomenon, in which a seemingly small change of the parameters causes a jump from provability to unprovability, here with respect to IΣ IΣ IΣ 1 .…”

“…We shall now consider more general sequences, where the entries no longer obey a particular formation law, but only satisfy some growth conditions defined in terms of Garside's complexity, in the spirit of the sentences considered by Friedman [19]. The main result here is that there exists a precise description of the conditions that lead from IΣ IΣ IΣ 1 -provability to IΣ IΣ IΣ 1 -unprovability, thus witnessing a quick phase transition phenomenon analogous to those investigated in [40][41][42].…”

“…What makes this possible is that these two functions cannot be distinguished inside IΣ IΣ IΣ 1 . The general idea of the proof, which is reminiscent of the analysis of phase transition for the Kruskal theorem [40][41][42][43], consists in starting with a long descending sequence, typically a G 3 -sequence (or, equivalently, any sequence witnessing for the principle WO ) and then constructing a new sequence by dilating the original one so as to lower the complexity of the entries. The argument requires that sufficiently many braids of low complexity be available, and this is where the estimate of Corollary 3.9 is crucial.…”

Classifying the phase transition threshold for Ackermannian functions

“…These results can be described intuitively as phase transitions from provability to unprovability of an assertion by varying a threshold parameter [13,16,17,21]. Another face of this phenomenon is the transition from slow-growing to fastgrowing computable functions [15,18].…”

Phase transition thresholds for some Friedman‐style independence results

“…These results can be described intuitively as phase transitions from provability to unprovability of an assertion by varying a threshold parameter [20,21,22]. Another face of this phenomenon is the transition from slow-growing to fast-growing computable functions [19,23].…”