A proof-theoretic analysis of collection
type: Item
instance of: scholarly article
title: A proof-theoretic analysis of collection
zbMATH Open document ID: 0916.03038
published in: Archive for Mathematical Logic
publication date: 1999-07-15T00:00:00Z
review text: The author investigates relations -- implication, independence, conservativeness, etc. -- among collection principles \((\text{B}\Sigma_n)\), induction axioms \((\text{I}\Sigma_n)\), and reflection principles \((\text{RFN}_T(\Pi_n))\). Besides producing new results, he gives new proofs to known results and answers open problems. For instance, Paris's result that \(\text{B}\Sigma_{n+ 1}\) is conservative over \(\text{I}\Sigma_n\) with respect to \(\Pi_{n+2}\) sentences is proved here by proof-theoretic means (cut-elimination, Herbrand theorem). \(\text{B}\Sigma_{n+ 1}\) is shown to be not derivable from all the true \(\Pi_{n+ 2}\) sentences; that answers a problem raised by Gaifman and Dimitracopoulos. There are a number of results about the equivalence of systems. For example, for \(k>n\), \(\text{B}\Sigma_n+ \Sigma_k\)-IR and \((\text{I}\Sigma_{n- 1})^{k+ 1}_\omega\) are equivalent. [`IR' stands for `induction rule', and for a theory \(T\), \((T)^n_{m+ 1}\) is \((T)^n_m+ \text{RFN}_{(T)^n_m}(\Pi_n)\).] The author provides a diagram that indicates relations about conservativeness and equivalence.
Mathematics Subject Classification ID: 03F30
Mathematics Subject Classification ID: 03F25
Mathematics Subject Classification ID: 03F05
Mathematics Subject Classification ID: 03D20
zbMATH DE Number: 1210618
zbMATH Keywords: collection principle
zbMATH Keywords: induction axiom
zbMATH Keywords: reflection principle
author: Lev D. Beklemishev
MaRDI profile type: Publication
full work available at URL: s001530050099
OpenAlex ID: W2091112568
DOI: 10.1007/S001530050099
description: scientific article; zbMATH DE number 1210618