Definability and almost disjoint families

TY - JOUR

T1 - Definability and almost disjoint families

AU - Törnquist, Asger Dag

PY - 2018/5/25

Y1 - 2018/5/25

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.R.D. Mathias in 1969. 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 ℵ1 L[a]<ℵ1, then there are no Σ2 1[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.R.D. Mathias in 1969. 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 ℵ1 L[a]<ℵ1, then there are no Σ2 1[a] infinite mad families.

U2 - 10.1016/j.aim.2018.03.005

DO - 10.1016/j.aim.2018.03.005

M3 - Journal article

SN - 0001-8708

VL - 330

SP - 61

EP - 73

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -