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Chang's conjecture

Chang’s conjecture is a model theoretic assertion which implies many structures of a certain variety have elementary substructures of another variety. Chang’s conjecture was originally formulated in 1963 by Chen Chung Chang and Vaught.

Chang’s conjecture is equiconsistent over $\text{ZFC}$ to the existence of the $\omega_1$-Erdős cardinal. In particular, if you collapse an $\omega_1$-Erdős cardinal to $\omega_2$ with the Silver collapse, then in the resulting model Chang’s conjecture holds. On the other hand, if Chang’s conjecture is true, then $\omega_2$ is $\omega_1$-Erdős in a transitive inner model of $\text{ZFC}$. (Donder & Levinski, 1989)

Chang’s conjecture implies $0^{\#}$ exists. (Kanamori, 2009)


The notation $(\kappa,\lambda)\twoheadrightarrow(\nu,\mu)$ is the assertion that every structure $\mathfrak{A}=(A;R^A…)$ with a countable language such that $|A|=\kappa$ and $|R^A|=\lambda$ has a proper elementary substructure $\mathfrak{B}=(B;R^B…)$ with $|B|=\nu$ and $|R^B|=\mu$.

This notation is somewhat intertwined with the square bracket partition properties. Namely, letting $\kappa\geq\lambda$ and $\kappa\geq\mu\geq\nu>\omega$, the partition property $\kappa\rightarrow[\mu]^{<\omega}_{\lambda,<\nu}$ is equivalent to the existence of some $\rho<\nu$ such that $(\kappa,\lambda)\twoheadrightarrow(\mu,\rho)$. (Kanamori, 2009)

As a result, some large cardinal axioms and partition properties can be described with this notation. In particular:

Chang’s conjecture is precisely $(\aleph_2,\aleph_1)\twoheadrightarrow(\aleph_1,\aleph_0)$. Chang’s conjecture is equivalent to the partition property $\omega_2\rightarrow[\omega_1]_{\aleph_1,<\aleph_1}^{<\omega}$. (Kanamori, 2009)


There are many stronger variants of Chang’s conjecture. Here are a few and their upper bounds for consistency strength (all can be found in (Eskrew & Hayut, 2016)):


  1. Donder, H.-D., & Levinski, J.-P. (1989). Some principles related to Changś conjecture. Annals of Pure and Applied Logic.
  2. Kanamori, A. (2009). The higher infinite (Second, p. xxii+536). Springer-Verlag.
  3. Jech, T. J. (2003). Set Theory (Third). Springer-Verlag.
  4. Eskrew, M., & Hayut, Y. (2016). On the consistency of local and global versions of Changś Conjecture.
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