Chang's conjecture

Template:Short description In model theory, a branch of mathematical logic, Chang's conjecture, attributed to Chen Chung Chang by Template:Harvtxt, 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 $({\omega }_2,{\omega }_1)\twoheadrightarrow ({\omega }_1,\omega )$.

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 a weak version of the reverse implication: if CC is not only consistent but actually 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 $({\omega }_3,{\omega }_2)\twoheadrightarrow ({\omega }_2,{\omega }_1)$ was shown by Laver from the consistency of a huge cardinal.

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