\(\Sigma _{1}\)-elementarity and Skolem hull operators
type: Item
instance of: scholarly article
title: \(\Sigma _{1}\)-elementarity and Skolem hull operators
zbMATH Open document ID: 1117.03064
author: Gunnar Wilken
published in: Annals of Pure and Applied Logic
publication date: 2007-02-14T00:00:00Z
review text: This article is reviewed together with the author's paper [``Ordinal arithmetic based on Skolem hulling'', Ann. Pure Appl. Logic 145, No. 2, 130--161 (2007; Zbl 1117.03063)] above. These two articles together with the earlier one [``The Bachmann-Howard structure in terms of \(\Sigma_1\)-elementarity'', Arch. Math. Logic 45, No. 7, 807--829 (2006; Zbl 1110.03053)] and the later one [``Assignment of ordinals to patterns of resemblance'', J. Symb. Log. 72, No. 2, 704--720 (2007; Zbl 1121.03081)] constitute an extensively revised version of the author's Dissertation [\(\Sigma_1\)-elementarity and Skolem hull operators. Münster: Univ. Münster, Fachbereich Mathematik und Informatik (2004; Zbl 1075.03028)]. In it, he develops and investigates ordinal notation systems based on two different approaches. The first approach is based on ``\(\Sigma_1\)-elementarity'', which T. J. Carlson started, and the second uses the more classical idea of ``Skolem hulling''. The earliest of the four papers applies the elementarity approach to the Bachmann-Howard ordinal. The first one of the two papers reviewed here, in the following referred to as [I], presents a toolkit for the construction of a notation system by means of Skolem hulling, and the remaining two papers offer comparison und mutual influences of the two approaches. Elementarity is a semantic notion: start from (the graph of) the ordinal addition \(+\) and the usual \(\leq\). Define a new relation \(\leq_1\) on ordinals by induction so that \[ \alpha \leq_1 \beta \quad \hbox{iff} \quad (\alpha; 0, +, \leq, \leq_1) \leq_{\Sigma_1} (\beta; 0, +, \leq, \leq_1). \] Here, \(\leq_{\Sigma_1}\) is the usual relation of \(\Sigma_1\) substructures. [\(\leq_1\) is a partial order coarser than \(\leq\), and has interesting properties. For instance, \(\varepsilon_0\) is the least \(\alpha\) such that \(\alpha \leq_1 \alpha + \alpha\). For any \(\alpha\), \(\{\beta \mid \alpha \leq_1 \beta \}\) is a closed interval. This interval may be endl…
Mathematics Subject Classification ID: 03F15
zbMATH DE Number: 5126403
zbMATH Keywords: proof theory
zbMATH Keywords: ordinal notations
zbMATH Keywords: \(\Sigma _{1}\)-elementarity
zbMATH Keywords: collapsing functions
zbMATH Keywords: patterns of resemblance
MaRDI profile type: Publication
full work available at URL: j.apal.2006.07.004
OpenAlex ID: W2018457617
cites work: Ordinal arithmetic and \(\Sigma_1\)-elementarity
cites work: Elementary patterns of resemblance
cites work: Q3024338
cites work: The Bachmann-Howard structure in terms of \(\Sigma_1\)-elementarity
cites work: Ordinal arithmetic based on Skolem hulling
cites work: Assignment of ordinals to patterns of resemblance
DOI: 10.1016/J.APAL.2006.07.004
description: scientific article; zbMATH DE number 5126403