Summable gaps

It is proved, under Martin’s Axiom, that all (ω1, ω1) gaps in P(N) / fin are indestructible in any forcing extension by a separable measure algebra. This naturally leads to a new type of gap, a summable gap. The results of these investigations have applications in Descriptive Set Theory. For example, it is shown that under Martin’s Axiom the Baire categoricity of all ∆3 non-∆ 1 3-complete sets of reals requires a weakly compact cardinal.