Some conservation results on weak König's lemma
type: Item
instance of: scholarly article
title: Some conservation results on weak König's lemma
zbMATH Open document ID: 1016.03064
DOI: 10.1016/S0168-0072(01)00121-X
author: Stephen G. Simpson
author: Kazuyuki Tanaka
author: Takeshi Yamazaki
published in: Annals of Pure and Applied Logic
publication date: 2002-12-02T00:00:00Z
review text: The main conservation theorem of this paper is that if \( \text{WKL}_0\) proves \(\forall X \exists ! Y \varphi (x,y)\) with \(\varphi\) arithmetical, then so does \( \text{RCA}_0\). This result extends some of the previous work of \textit{T. Yamazaki} [Math. Log. Q. 46, No. 1, 105-110 (2000; Zbl 0942.03061)] and \textit{A. M. Fernandes} [Arch. Math. Log. 41, No. 1, 55-63 (2002)]. The proof technique utilizes forcing and the fact that for any countable model \(M\) of \(\text{RCA}_0\), there is a countable model \(M^\prime\) of \(\text{WKL}_0\) such that \(M\) and \(M^\prime\) have the same integers and the intersection of their set universes consists exactly of the \(M\)-recursive subsets of the integers of \(M\). For a more extensive treatment of advances in models of \(\text{WKL}_0\), see \textit{S. G. Simpson}'s ``\(\Pi^0_1\) sets and models of \(\text{WKL}_0\)'', which will appear in the forthcoming volume of collected articles on reverse mathematics which he is editing.
Mathematics Subject Classification ID: 03F35
Mathematics Subject Classification ID: 03B30
Mathematics Subject Classification ID: 03C25
Mathematics Subject Classification ID: 03D80
zbMATH DE Number: 1838867
zbMATH Keywords: RCA
zbMATH Keywords: WKL
zbMATH Keywords: hard core theorem
zbMATH Keywords: productive functions
zbMATH Keywords: forcing
zbMATH Keywords: universal trees
zbMATH Keywords: genericity
zbMATH Keywords: pointed perfect trees
reviewed by: Jeffry L. Hirst
MaRDI profile type: Publication
Wikidata QID: Q124849533
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full work available at URL: s0168-0072(01)00121-x
OpenAlex ID: W2134006590
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description: scientific article; zbMATH DE number 1838867