Patterns of resemblance of order 2
type: Item
instance of: scholarly article
title: Patterns of resemblance of order 2
zbMATH Open document ID: 1206.03050
author: Timothy J. Carlson
published in: Annals of Pure and Applied Logic
publication date: 2009-04-09T00:00:00Z
review text: This article continues the project of the author into developing ordinal notation systems based on \(\Sigma_n\)-elementary substructures. The basic idea is to consider finite substructures of the structures \((\text{ORD},0,+,\leq,\leq_1 ,\dots, \leq_n)\), where \(\leq_i\) is interpreted as \[ \alpha \leq_i \beta \iff (\alpha,0,+,\leq,\leq_1, \leq_n) \prec_{\Sigma_i} (\alpha,0,+,\leq,\leq_1 ,\dots, \leq_n). \] This idea is then varied. Here the operations are considered as partial operations. Let \(B=(A,0,+,\leq,\leq_1 ,\dots, \leq_n)\) be a minimal final substructure w.r.t. pointwise ordering. This structure can be described finally. Then \(B\) together with an \(a \in A\) can be used in order to denote the ordinal by which \(a\) is interpreted. This way, one obtains notations for ordinals. The set of ordinals which can be denoted this way forms the core of the structure. The goal of this project is to derive computable well-orderings based on this idea and to find an ultrafine structure for large-cardinal axioms based on such embeddings. Previous work as part of this project has influenced the development of ordinal notation systems of strength \(\Pi^1_2\)-CA and beyond by Michael Rathjen. This work can be seen as well as a project on developing strong natural ordinal notation systems, which originate from purely structural set-theoretic investigations rather than from defining an ordinal notation systems based on functions (which don't have a direct justification of being natural and therefore can always be varied in order to obtain quite different notation systems) -- however, the author doesn't make any claims in this direction, he does not even use the term ``natural well-ordering''. In this article the author studies patterns of resemblance of order 2, i.e.~\(n\) as above is \(2\). He computes the core of these patterns of resemblance and shows that it is a computable structure. The structure of the article is as follows: The paper presents first (Section~1) s…
reviewed by: Anton Setzer
Mathematics Subject Classification ID: 03F15
Mathematics Subject Classification ID: 03E10
Mathematics Subject Classification ID: 03E55
zbMATH DE Number: 5542408
zbMATH Keywords: ordinal notation systems
zbMATH Keywords: natural well-orderings
zbMATH Keywords: patterns resemblance
zbMATH Keywords: reflection principles
zbMATH Keywords: \(\Sigma_n\)-elementary substructures
zbMATH Keywords: proof theory
MaRDI profile type: Publication
full work available at URL: j.apal.2008.10.006
OpenAlex ID: W1987053402
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DOI: 10.1016/J.APAL.2008.10.006
description: scientific article; zbMATH DE number 5542408