Chang's conjecture

{{Short description|Mathematical conjecture}}

In model theory, a branch of mathematical logic, Chang's conjecture, attributed to Chen Chung Chang by {{harvtxt|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 $(\backslash omega\_2,\backslash omega\_1)\backslash twoheadrightarrow(\backslash omega\_1,\backslash omega)$.

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 $(\backslash omega\_3,\backslash omega\_2)\backslash twoheadrightarrow(\backslash omega\_2,\backslash omega\_1)$ was shown by Laver from the consistency of a huge cardinal.