Chang's conjecture

Statement

In model theory, a branch of mathematical logic, Chang's conjecture, attributed to Chen Chung Chang by (Vaught 1963, 309), states that every model of type (ω₂,ω₁) for a countable language has an elementary submodel of type (ω₁,ω). A model is of type (α,β) if it is of cardinality α and a unary relation is represented by a subset of cardinality β. The usual notation is $(\omega_2,\omega_1)\twoheadrightarrow(\omega_1,\omega)$.