Chang's conjecture
In model theory, a branch of mathematical logic, Chang's conjecture, attributed to Chen Chung Chang by, states that every model of type for a countable language has an elementary submodel of type. A model is of type if it is of cardinality α and a unary relation is represented by a subset of cardinality β. The usual notation is.
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, of cardinals is the claim
that every model of type for a countable language has an elementary submodel of type.
The consistency of was shown by Laver from the consistency of a huge cardinal.
