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 (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 ( ω 2 , ω 1 ) ↠ ( ω 1 , ω ) .

The axiom of constructibility implies that Chang's conjecture fails. Silver proved the consistency of Chang's conjecture from the consistency of an ω₁-Erdős cardinal. Hans-Dieter Donder showed the reverse implication: if CC holds, then ω₂ is ω₁-Erdős in K.

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 ( ω 3 , ω 2 ) ↠ ( ω 2 , ω 1 ) was shown by Laver from the consistency of a huge cardinal.