Chang's conjecture

• Definition
• Variants

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)

Definition

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:

• $\kappa$ is Rowbottom if and only if $\kappa>\aleph_1$ and for any uncountable $\lambda<\kappa$, $(\kappa,\lambda)\twoheadrightarrow(\kappa,\aleph_0)$. (Jech, 2003)
• $\kappa$ is Jónsson if and only if for any $\lambda\leq\kappa$, there is some $\nu\leq\kappa$ such that $(\kappa,\lambda)\twoheadrightarrow(\kappa,\nu)$. (Jech, 2003)

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)

Variants

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)):

• Assuming the consistency of a $\kappa$ which is $\kappa^{++}$-supercompact, it is consistent that there is a proper class of cardinals $\lambda$ such that $(\lambda^{+++},\lambda^{++})\twoheadrightarrow(\lambda^+,\lambda)$.
• Assuming the consistency of a $\kappa$ which is $\kappa^{++}$-supercompact, it is consistent that there is a proper class of cardinals $\kappa$ such that $(\lambda^{+\omega+2},\lambda^{+\omega+1})\twoheadrightarrow(\lambda^+,\lambda)$.
• Assuming the consistency of a cardinal $\kappa$ which is $\kappa^{+\omega+1}$-supercompact, it is consistent that $(\aleph_{\omega+1},\aleph_\omega)\twoheadrightarrow(\aleph_1,\aleph_0)$.
• Assuming the consistency of a huge cardinal, it is consistent that $(\kappa^{++},\kappa^+)\twoheadrightarrow(\mu^+,\mu)$ for every $\kappa$ and $\mu<\kappa^+$.
• It is unknown whether or not it is consistent that $(\aleph_{\omega_1+1},\aleph_{\omega_1})\twoheadrightarrow(\aleph_{\omega+1},\aleph_\omega)$.

References

1. Donder, H.-D., & Levinski, J.-P. (1989). Some principles related to Changś conjecture. Annals of Pure and Applied Logic. https://doi.org/10.1016/0168-0072(89)90030-4
2. Kanamori, A. (2009). The higher infinite (Second, p. xxii+536). Springer-Verlag. https://link.springer.com/book/10.1007%2F978-3-540-88867-3
3. Jech, T. J. (2003). Set Theory (Third). Springer-Verlag. https://logic.wikischolars.columbia.edu/file/view/Jech%2C+T.+J.+%282003%29.+Set+Theory+%28The+3rd+millennium+ed.%29.pdf
4. Eskrew, M., & Hayut, Y. (2016). On the consistency of local and global versions of Changś Conjecture.

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