Model theory without choice? Categoricity

Abstract

We prove łos conjecture=Morley theorem in ZF, with the same characterization, i.e., of first order countable theories categorical in ℵ_α for some (equivalently for every ordinal) α > 0. Another central result here in this context is: the number of models of a countable first order T of cardinality ℵ_α is either ≥ |α| for every α or it has a small upper bound (independent of α close to ℶ₂).