Definability and almost disjoint families
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Definability and almost disjoint families. / Törnquist, Asger Dag.
I: Advances in Mathematics, Bind 330, 2018, s. 61-73.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Definability and almost disjoint families
AU - Törnquist, Asger Dag
PY - 2018
Y1 - 2018
N2 - We show that there are no infinite maximal almost disjoint ("mad") families in Solovay's model, thus solving a long-standing problem posed by A.D.R. Mathias in 1967. We also give a new proof of Mathias' theorem that no analytic infinite almost disjoint family can be maximal, and show more generally that if Martin's Axiom holds at κ<2ℵ0, then no κ-Souslin infinite almost disjoint family can be maximal. Finally we show that if ℵL[a]1<ℵ1, then there are no Σ12[a] infinite mad families.
AB - We show that there are no infinite maximal almost disjoint ("mad") families in Solovay's model, thus solving a long-standing problem posed by A.D.R. Mathias in 1967. We also give a new proof of Mathias' theorem that no analytic infinite almost disjoint family can be maximal, and show more generally that if Martin's Axiom holds at κ<2ℵ0, then no κ-Souslin infinite almost disjoint family can be maximal. Finally we show that if ℵL[a]1<ℵ1, then there are no Σ12[a] infinite mad families.
U2 - 10.1016/j.aim.2018.03.005
DO - 10.1016/j.aim.2018.03.005
M3 - Journal article
VL - 330
SP - 61
EP - 73
JO - Advances in Mathematics
JF - Advances in Mathematics
SN - 0001-8708
ER -
ID: 184033378