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 . We show that the resulting slow growing hierarchy is eventually dominated by a fixed elementary recursive function if is equal to zero. We show further that the resulting slow growing hierarchy exhausts the provably recursive functions of ID1 if 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 . 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 function...

• Type: Journal
• Year: 2006
• Where: MSCS
• Authors: Andreas Weiermann
• Keywords: Bachmann Ordinal, Fundamental Sequences, MSCS 2006, Recursive Functions