Trivial automorphisms

Abstract

We prove that the statement ‘For all Borel ideals I and J on ω, every isomorphism between Boolean algebras P(ω)/I and P(ω)/J has a continuous representation’ is relatively consistent with ZFC. In this model every isomorphism between P(ω)/I and any other quotient P(ω)/J over a Borel ideal is trivial for a number of Borel ideals I on ω.We can also assure that the dominating number, σ, is equal to ℵ1 and that (Formula presented.). Therefore, the Calkin algebra has outer automorphisms while all automorphisms of P(ω)/Fin are trivial.Proofs rely on delicate analysis of names for reals in a countable support iteration of Suslin proper forcings.