TY - JOUR
T1 - The spectrum of independence
AU - Fischer, Vera
AU - Shelah, Saharon
N1 - Publisher Copyright:
© 2019, The Author(s).
PY - 2019/11
Y1 - 2019/11
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
UR - https://www.scopus.com/pages/publications/85062149689
U2 - 10.1007/s00153-019-00665-y
DO - 10.1007/s00153-019-00665-y
M3 - Article
SN - 0933-5846
VL - 58
SP - 877
EP - 884
JO - Archive for Mathematical Logic
JF - Archive for Mathematical Logic
IS - 7-8
ER -