Ordinal Notations in Caucal Hierarchy

TL;DR

This paper explores the properties of well-orderings within the Caucal hierarchy, demonstrating their extension with monadically definable cofinal sequences and analyzing the impact on fast-growing hierarchies.

Contribution

It introduces monadically definable cofinal sequences with Bachmann property for well-orderings in the Caucal hierarchy and compares growth rates of related hierarchies.

Findings

Well-orderings in Caucal hierarchy have order types less than ε₀.

Any such well-ordering can be extended with monadically definable cofinal sequences.

Hierarchies based on these notations align with L"ob-Wainer hierarchy for ordinals less than ω^ω.

Abstract

Caucal hierarchy is a well-known class of graphs with decidable monadic theories. It were proved by L. Braud and A. Carayol that well-orderings in the hierarchy are the well-orderings with order types less than $\varepsilon_0$. Naturally, every well-ordering from the hierarchy could be considered as a constructive system of ordinal notations. In proof theory constructive systems of ordinal notations with fixed systems of cofinal sequences are used for the purposes of classification of provable recursive functions of theories. We show that any well-ordering from the hierarchy could be extended by a monadically definable system of cofinal sequences with Bachmann property. We show that the growth speed of functions from fast-growing hierarchy based on constructive ordinal notations from Caucal hierarchy may be only slightly influenced by the choice of monadically definable systems of cofinal sequences. We show that for ordinals less than $\omega^\omega$ a fast-growing hierarchy based on any system of ordinal notations from Caucal hierarchy coincides with L\"ob-Wainer hierarchy.