Rekursion über Dilatoren und die Bachmann-Hierarchie

Summary

A hierarchy (J ^g _D ) _{D Dilator} of ordinal functionsJ ^g _D : On→On is introduced and studied. It is a hierarchy of iterations relative to some giveng:OnarOn, defined by primitive recursion on dilators. This hierarchy is related to a Bachmann hierarchy$\left( {\phi _\alpha ^g } \right)_{\alpha< \varepsilon _{\Omega {\mathbf{ }} + {\mathbf{ }}1} }$, which is built on an iteration ofg ↑ Ω as initial function.

This Bachmann hierarchy$\left( {\phi _\alpha ^g } \right)_{\alpha< \varepsilon _{\Omega {\mathbf{ }} + {\mathbf{ }}1} }$ itself is shown to be explicitly definable by

$$\phi _\alpha ^g (\eta ) = g^{\Omega ^\alpha \cdot (1 + \eta )} (0)$$

from a hierarchy of iterationsg ^α :Ω→Ω for weakly monotonicg: Ω→Ω withg(0)>0. Forg=λη. (1+η) ·ω org=λη. 1, in particular, one obtains as$\left( {\phi _\alpha ^g } \right)_{\alpha< \varepsilon _{\Omega {\mathbf{ }} + {\mathbf{ }}1} }$ Bachmann hierarchy.

With every ordinalα≦ε _Ω+1 a dilatorD _α is associated.D _α is “below” the dilator$\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{2} + Id} \right)_{(\omega )}$, which is defined as$\mathop {\sup }\limits_{n< \omega } (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{2} + Id)_{(n)}$ with (¯2+Id)₍₀₎:=1 and$(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{2} + Id)_{(n + 1)} : = (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{2} + Id)^{(2 + Id)_{(n)} }$. For weakly monotonicg:OnarOn satisfyingg(0)>0 and$g(\Omega ) \subseteqq \Omega$, and forη<Ω andα<ε _Ω+1 it is proved that

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