Kripke-Platek set theory and the anti-foundation axiom
type: Item
instance of: scholarly article
zbMATH Open document ID: 1002.03050
author: Michael Rathjen
publication date: 2003-01-13T00:00:00Z
Mathematics Subject Classification ID: 03F25
Mathematics Subject Classification ID: 03F15
Mathematics Subject Classification ID: 03F35
Mathematics Subject Classification ID: 03E70
Mathematics Subject Classification ID: 03E30
zbMATH DE Number: 1694860
zbMATH Keywords: Kripke-Platek set theory
zbMATH Keywords: anti-foundation axiom
zbMATH Keywords: second-order arithmetic
zbMATH Keywords: proof-theoretic strength
zbMATH Keywords: decoration
zbMATH Keywords: graphs
MaRDI profile type: Publication
OpenAlex ID: W2007535202
DOI: 10.1002/1521-3870(200111)47:4%3C435::AID-MALQ435%3E3.0.CO;2-7
full work available at URL: https://doi.org/10.1002/1521-3870(200111)47:4%3C435::aid-malq435%3E3.0.co
title: Kripke-Platek set theory and the anti-foundation axiom
published in: Mathematical Logic Quarterly (MLQ)
review text: KPA is the label the author attaches to the theory he considers, namely, the one obtained from Kripke-Platek set theory by replacing the foundation axiom by the anti-foundation axiom, AFA. He shows that (i) KPA and a second-order arithmetic \(\Delta^1_2\text{-CA}_0\) have the same proof-theoretic strength, and (ii) so do KPA+IND\(_\omega\) and \(\Delta^1_2\)-CA, by interpreting one theory into the other. AFA is stated in terms of graph theory. Given a directed graph, a decoration is, by definition, an assignment \(d\) of sets to nodes such that \(d(x)=\{d(y) \mid\exists\) edge from \(x\) to \(y\}\). AFA dictates that a graph has a unique decoration. The author considers graphs based on linear orders on \(\omega\), and shows their being well-ordered is \(\Delta_1\) in KPA.
description: scientific article; zbMATH DE number 1694860