# Chang's conjecture

In model theory, a branch of mathematical logic, Chang's conjecture, attributed to Chen Chung Chang by Vaught (1963, p. 309), states that every model of type (ω21) 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 ${\displaystyle (\omega _{2},\omega _{1})\twoheadrightarrow (\omega _{1},\omega )}$.
More generally, Chang's conjecture for two pairs (α,β), (γ,δ) of cardinals is the claim that every model of type (α,β) for a countable language has an elementary submodel of type (γ,δ). The consistency of ${\displaystyle (\omega _{3},\omega _{2})\twoheadrightarrow (\omega _{2},\omega _{1})}$ was shown by Laver from the consistency of a huge cardinal.