On oracle-c.c. and “P(ω)/ finite has no non-trivial automorphism”

Abstract

Here we present how to iterate forcing satisfying the $\bar M$-c.c. for an oracle $\bar M$. This enables us, in some sense, to “construct” the set of reals in ω ₂ steps with omitting types along the way (i.e., imitating constructions of model theory). We use it to prove the consistency of “P(ω) /finite has no non-trivial automorphism”, which is equivalent to “β(N) −N has only trivial homeomorphisms”.