The spectrum of independence

Vera Fischer; Saharon Shelah

doi:10.1007/s00153-019-00665-y

The spectrum of independence

Vera Fischer, Saharon Shelah

Department of Mathematics

Publications: Contribution to journal › Article › Peer Reviewed

Abstract

We study the set of possible sizes of maximal independent families to which we refer as spectrum of independence and denote Spec (mif). Here mif abbreviates maximal independent family. We show that:1.whenever κ ₁< ⋯ < κ _n are finitely many regular uncountable cardinals, it is consistent that {κi}i=1n⊆Spec(mif);2.whenever κ has uncountable cofinality, it is consistent that Spec (mif) = { ℵ ₁, κ= c}. Assuming large cardinals, in addition to (1) above, we can provide that (κi,κi+1)∩Spec(mif)=∅for each i, 1 ≤ i< n.

Original language	English
Pages (from-to)	877–884
Number of pages	8
Journal	Archive for Mathematical Logic
Volume	58
Issue number	7-8
DOIs	https://doi.org/10.1007/s00153-019-00665-y
Publication status	Published - Nov 2019

Austrian Fields of Science 2012

101013 Mathematical logic

Keywords

Cardinal characteristics
Independent families
Sacks indestructibility
Spectrum
Ultrapowers

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title = "The spectrum of independence",

abstract = "We study the set of possible sizes of maximal independent families to which we refer as spectrum of independence and denote Spec (mif). Here mif abbreviates maximal independent family. We show that:1.whenever κ 1< ⋯ < κ n are finitely many regular uncountable cardinals, it is consistent that \{κi\}i=1n⊆Spec(mif);2.whenever κ has uncountable cofinality, it is consistent that Spec (mif) = \{ ℵ 1, κ= c\}. Assuming large cardinals, in addition to (1) above, we can provide that (κi,κi+1)∩Spec(mif)=∅for each i, 1 ≤ i< n. ",

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AU - Shelah, Saharon

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N2 - We study the set of possible sizes of maximal independent families to which we refer as spectrum of independence and denote Spec (mif). Here mif abbreviates maximal independent family. We show that:1.whenever κ 1< ⋯ < κ n are finitely many regular uncountable cardinals, it is consistent that {κi}i=1n⊆Spec(mif);2.whenever κ has uncountable cofinality, it is consistent that Spec (mif) = { ℵ 1, κ= c}. Assuming large cardinals, in addition to (1) above, we can provide that (κi,κi+1)∩Spec(mif)=∅for each i, 1 ≤ i< n.

AB - We study the set of possible sizes of maximal independent families to which we refer as spectrum of independence and denote Spec (mif). Here mif abbreviates maximal independent family. We show that:1.whenever κ 1< ⋯ < κ n are finitely many regular uncountable cardinals, it is consistent that {κi}i=1n⊆Spec(mif);2.whenever κ has uncountable cofinality, it is consistent that Spec (mif) = { ℵ 1, κ= c}. Assuming large cardinals, in addition to (1) above, we can provide that (κi,κi+1)∩Spec(mif)=∅for each i, 1 ≤ i< n.

KW - Cardinal characteristics

KW - Independent families

KW - Sacks indestructibility

KW - Spectrum

KW - Ultrapowers

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DO - 10.1007/s00153-019-00665-y

M3 - Article

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SP - 877

EP - 884

JO - Archive for Mathematical Logic

JF - Archive for Mathematical Logic

IS - 7-8

ER -

The spectrum of independence

Abstract

Austrian Fields of Science 2012

Keywords

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