On the strength of the finite intersection principle
type: Item
instance of: scholarly article
title: On the strength of the finite intersection principle
zbMATH Open document ID: 1302.03035
author: Damir D. Dzhafarov
author: Carl Mummert
published in: Israel Journal of Mathematics
publication date: 2013-11-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/1109.3374
review text: Consider the following (intersection) properties of families of sets \(A\). The \(D_k\) intersection property is the property that the intersection of each \(k\) distinct sets of \(A\) is empty. The \(\overline{D}_k\) intersection property is the property that the intersection of each \(k\) sets of \(A\) is non-empty. The \(F\) intersection property is the property that each finite intersection of sets of \(A\) is non-empty. It is a classical result that for each of the above intersection properties \(P\) the statement that each family \(A\) of sets contains a maximal subfamily \(A'\subseteq A\) satisfying \(P\), is equivalent to the axiom of choice over ZF. In this paper, these statements are analyzed in the fashion of reverse mathematics. To formulate them in the language of reverse mathematics the authors restrict their attention to countable families of sets. Writing \(P\mathsf{IP}\) for this statement with property \(P\), the following is obtained. \(D_k\mathsf{IP}\) is equivalent to \(\mathsf{ACA}_0\), \(F\mathsf{IP} \rightarrow \overline{D}_{k+1}\mathsf{IP} \rightarrow \overline{D}_k\mathsf{IP}\), and the \(\Pi^0_1\)-generic principle implies \(F\mathsf{IP}\) and \(\mathsf{WKL}_0\nvdash \overline{D}_2\mathsf{IP}\). Moreover, connections to the atomic model theorem and the omitting types theorem from [\textit{D. R. Hirschfeldt} et al., Trans. Am. Math. Soc. 361, No. 11, 5805--5837 (2009; Zbl 1184.03005)] are given.
reviewed by: Alexander P. Kreuzer
Mathematics Subject Classification ID: 03B30
Mathematics Subject Classification ID: 03E25
Mathematics Subject Classification ID: 03F35
zbMATH DE Number: 6221778
zbMATH Keywords: reverse mathematics
zbMATH Keywords: intersection principles
zbMATH Keywords: axiom of choice
MaRDI profile type: Publication
OpenAlex ID: W3099017129
Wikidata QID: Q57951143
arXiv ID: 1109.3374
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DOI: 10.1007/S11856-012-0150-9
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description: scientific article; zbMATH DE number 6221778