Working in applications of logic to operator algebras, we show how working in different axiomatic systems has an impact on the structure of isomorphisms and embeddings of corona C*-algebras. On one hand, the assumption of the Continuum Hypothesis allows us to show, among other things, that the homeomorphism group of a Stone-Cech remainder of a locally compact noncompact manifold is large and wild. On the other hand, assuming strong versions of the Baire Category theorem negating the Continuum Hypothesis, we prove a strong version of lifting theorem allowing us to provide rigidity results for isomorphisms and embeddings of certain corona C*-algebras. We also provide some stability results for maps between C*-algebras with are absolute among set theory and have connections with perturbation theory for operator algebras.