A comparison of two systems of ordinal notations
In this article the author compares the Bachmann-Howard ordinal notation system with ordinals definable by using higher-order fixed point operators on the class ordinals. The latter system was extracted from Howard's paper [\textit{W. A. Howard}, ``A system of abstract constructive ordinals'', J. Symb. Log. 37, 355--374 (1972; Zbl 0264.02026)], which is closely related to Aczel's article [\textit{P. Aczel}, ``Describing ordinals using functionals of transfinite type'', J. Symb. Log. 37, 35--47 (1972; Zbl 0264.02025)]. The precise statement shown is as follows: Let \({\mathbb O}\text{rd}^{(0)}\) be the class of ordinals, and define \({\mathbb O}\text{rd}^{(l)}\) inductively by \({\mathbb O}\text{rd}^{(l+1)}:= {\mathbb O}\text{rd}^{(l)} \rightarrow {\mathbb O}\text{rd}^{(l)}\). Furthermore the author mentions subsets \({\mathbb H}^{(l)}\) (``helpful functions'') of \({\mathbb O}\text{rd}^{(l)}\), which are not defined in this paper but in some notes available from the author's webpage. Let \(\mathbf N \in {\mathbb H}^{(1)}\) be fixed, \(\epsilon\) a fixed point of \(\lambda \alpha.\omega^\alpha\). The author introduces ordinals \(\Delta[l]\) and \(\nabla[l]\) for \(l \in \omega\) as follows: Let \(\text{\textbf{Fix}} \in {\mathbb O}\text{rd}^{(2)}\), \(\text{\textbf{Fix}}f\zeta:= f^\omega(\zeta +1)\), which is for \(f\) normal \(\min \{ \zeta < \nu \mid \nu = f\nu \}\). Let, for \(l \in \omega\), \([l]: {\mathbb O}\text{rd}^{(l+2)}\) be defined by \([l]fg_l\cdots g_1:= \text{\textbf{Fix}}(\lambda \alpha: {\mathbb O}\text{rd}.f^\alpha g_l\cdots g_1)\). Let \(\Delta[0]:= \epsilon\), \(\Delta[1]:= \mathbf N \epsilon\), \(\Delta[l+2]:= [l] \cdots [0]\mathbf N\epsilon \). Let \(\psi\) be a suitable variant of the standard collapsing function based on \(\epsilon\) and \(\mathbf N\) (especially \(\psi(0) = \varepsilon\) and \(\psi(\alpha)\) is always a fixed-point of \(\mathbf N\)). Let \(\nabla[l]= \psi(\Omega_l(0))\) with \(\Omega_0(0):= 0\), \(\Omega_{l+1}(0):= \Omega^{\Omega_l(0)}\). Then the result obtained in this paper is that \(\Delta[l] = \nabla[l]\). In fact a full translation between ordinals \(\psi(\xi)\) and ordinals defined in the author's algebra of fixed point functions is given. According to the author, the system based on fixed point operators is from below, whereas the ordinal notation system based on the collapsing function is not. Although it is clear that the system based on fixed point operators is an improvement w.r.t. the system by Bachmann, there is no precise statement of what the author means by ``from below'' -- it seems that he means some approach based on forming fixed points of ordinal functions. It would be very interesting if a precise definition could be given. An alternative approach towards defining what is meant by ``from below'' has been given by the reviewer [\textit{A. Setzer}, ``Ordinal systems'', in: S. B. Cooper et al. (eds.), Sets and proofs. Invited papers from the Logic colloquium '97, Lond. Math. Soc. Lect. Note Ser. 258, 301--338 (1999; Zbl 0944.03054)]. This approach to reaching high ordinals by using fixed points of hyper type operators is a very fascinating approach, which is quite different from the approach usually taken in the development of ordinal notation systems. We believe that this is an interesting contribution towards the question, what is meant by natural ordinal notation systems and ordinal notation systems from below (not that we believe that one will ever find a satisfactory notion of a natural ordinal notation system). The article is written in a very interesting and enjoyable style, and it is fun to read it. However, this article doesn't give full details, even the important definition of \({\mathbb H}^{(l)}\) (which could be given easily) is missing. Many proof details are given in unpublished papers available from the author's homepage. Therefore, in order to understand the proofs fully, one has to read through these (rather long) notes. We hope that some of this material will be published soon, maybe in a more condensed form than those unpublished notes.
scientific article; zbMATH DE number 2101703