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](Erdos.md "Erdos") 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 $\text{ZFC}$. {% cite Donder1989 %} Chang's conjecture implies $0^{\sharp}$ exists. {% cite Kanamori2009 %} ## 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 [](Elementary%20embedding.md) $\mathfrak{B}=(B;R^B...)$ with $\|B\|=\nu$ and $\|R^B\|=\mu$. This notation is somewhat intertwined with the [](Partition%20property.md). 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 $(\kappa,\lambda)\twoheadrightarrow(\mu,\rho)$. {% cite Kanamori2009 %} As a result, some large cardinal axioms and partition properties can be described with this notation. In particular: - $\kappa$ is [Rowbottom](Rowbottom.md "Rowbottom") if and only if $\kappa>\aleph_1$ and for any uncountable $\lambda<\kappa$, $(\kappa,\lambda)\twoheadrightarrow(\kappa,\aleph_0)$. {% cite Jech2003 %} - $\kappa$ is [Jónsson](Jonsson.md "Jonsson") if and only if for any $\lambda\leq\kappa$, there is some $\nu\leq\kappa$ such that $(\kappa,\lambda)\twoheadrightarrow(\kappa,\nu)$. {% cite Jech2003 %} **Chang's conjecture** is precisely $(\aleph_2,\aleph_1)\twoheadrightarrow(\aleph_1,\aleph_0)$. Chang's conjecture is equivalent to the [](Partition%20property.md) $\omega_2\rightarrow\[\omega_1\]_{\aleph_1,<\aleph_1}^{<\omega}$. {% cite Kanamori2009 %} ## 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 {% cite Eskrew2016 %}): - Assuming the consistency of a $\kappa$ which is [$\kappa^{++}$-supercompact](Supercompact.md "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](Supercompact.md "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](Supercompact.md "Supercompact"), it is consistent that $(\aleph_{\omega+1},\aleph_\omega)\twoheadrightarrow(\aleph_1,\aleph_0)$. - Assuming the consistency of a [huge](Huge.md "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 $(\aleph_{\omega_1+1},\aleph_{\omega_1})\twoheadrightarrow(\aleph_{\omega+1},\aleph_\omega)$.