Chang's conjecture

In model theory, a branch of mathematical logic, Chang's conjecture, attributed to Chen Chung Chang by (Vaught 1963), 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 [math]\displaystyle{ (\omega_2,\omega_1)\twoheadrightarrow(\omega_1,\omega) }[/math]. 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 [math]\displaystyle{ (\omega_3,\omega_2)\twoheadrightarrow(\omega_2,\omega_1) }[/math] was shown by Laver from the consistency of a huge cardinal.

References

• Chang, Chen Chung; Keisler, H. Jerome (1990), Model Theory, Studies in Logic and the Foundations of Mathematics (3rd ed.), Elsevier, ISBN 978-0-444-88054-3 
• Vaught, R. L. (1963), "Models of complete theories", Bulletin of the American Mathematical Society 69: 299–313, doi:10.1090/S0002-9904-1963-10903-9, ISSN 0002-9904, https://www.ams.org/bull/1963-69-03/S0002-9904-1963-10903-9/home.html