Ausgezeichnete Folgen Für Prädikative Ordinalzahlen und Prädikativ‐Rekursive Funktionen

Some interesting connections between the slow growing hierarchy and the Ackermann function

“…This situation is in full accordance to Cichon's principle since for a certain natural tree ordinal representation for To we have that the functions, which are elementary recursive in the Ackermann function, are classified in terms of the slow growing hierarchy along To. (See, for example, Vogel (1977) [22] for a proof of this folklore result. )…”

“…Recall that for a given assignment •[•]: (eo n Lim) xco -> co the slow growing hierarchy (G a ) a<60 is defined as follows: Go(x) := 0, G a+ \(x) := G a {x) + 1 and Gx{x) := Gx[ X ]{x). See, for example, Vogel (1977) [22] or Girard (1981) [13] for a definition.) Surprisingly, it turns out that the slow growing hierarchy (G a ) a<£0 -when it is defined with respect to •[•]() and -[-]2 -consists of elementary recursive functions only whereas (G a ) a<eo -when it is defined with respect to []i -is fast growing, i.e.…”

“…Eventually, for proving the hierarchy comparison theorem for To we introduce a new method for counting fixed points. This is reflected by an appropriate Grzegorczyk hierarchy of primitive recursive functions (Xx, y, z.F m (x, y, z)) m<m whose definition is partly inspired from corresponding definitions given by Vogel (1977) in [22] and by Cichon and Wainer (1983) in [9]. Nevertheless the concrete definition of the hierarchy {Xx, y, z.F m {x, y, z)) m<aj has been extracted from tedious, lengthy and involved numerical evaluations of the slow growing functions involved.…”