Forcing indestructibility of MAD families

Abstract

Let A ⊆ [ ω ] ω be a maximal almost disjoint family and assume P is a forcing notion. Say A is P -indestructible if A is still maximal in any P -generic extension. We investigate P -indestructibility for several classical forcing notions P . In particular, we provide a combinatorial characterization of P -indestructibility and, assuming a fragment of MA, we construct maximal almost disjoint families which are P -indestructible yet Q -destructible for several pairs of forcing notions ( P , Q ) . We close with a detailed investigation of iterated Sacks indestructibility.