Knight's model, its automorphism group, and characterizing the uncountable cardinals

摘要

We show that every N_α (α < ω_1) can be characterized by the Scott sentence of some countable model; moreover there is a countable structure whose Scott sentence of some countable model; moreover there is a countable structure whose Scott sentence characterizes N_1 but whose automorphism group fails the topological Vaught conjecture on analytic sets. We obtain some partial information on Ulm type dichotomy theorems for the automorphism group of Knight's model.