A comparison of two systems of ordinal notations
type: Item
instance of: scholarly article
title: A comparison of two systems of ordinal notations
zbMATH Open document ID: 1067.03063
author: Harold Simmons
published in: Archive for Mathematical Logic
publication date: 2004-09-22T00:00:00Z
review text: 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^{\O…
reviewed by: Anton Setzer
Mathematics Subject Classification ID: 03F15
Mathematics Subject Classification ID: 03F03
zbMATH DE Number: 2101703
zbMATH Keywords: ordinal notation systems
zbMATH Keywords: Gödel's \(T\)
zbMATH Keywords: \({\roman{ID}}_1\)
zbMATH Keywords: Bachmann-Howard ordinal
zbMATH Keywords: ordinal notation systems from below
zbMATH Keywords: fixed points
zbMATH Keywords: normal functions
zbMATH Keywords: critical ordinals
zbMATH Keywords: Veblen function
MaRDI profile type: Publication
full work available at URL: s00153-003-0177-z
OpenAlex ID: W2014501253
DOI: 10.1007/S00153-003-0177-Z
description: scientific article; zbMATH DE number 2101703