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 $\text{ZFC}$. [1]
Chang's conjecture implies $0^{\#}$ exists. [2]
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 $(\kappa,\lambda)\twoheadrightarrow(\mu,\rho)$. [2]
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 $\lambda<\kappa$, $(\kappa,\lambda)\twoheadrightarrow(\kappa,\aleph_0)$. [3]
- $\kappa$ is Jónsson if and only if for any $\lambda\leq\kappa$, there is some $\nu\leq\kappa$ such that $(\kappa,\lambda)\twoheadrightarrow(\kappa,\nu)$. [3]
Chang's conjecture is precisely $(\aleph_2,\aleph_1)\twoheadrightarrow(\aleph_1,\aleph_0)$. Chang's conjecture is equivalent to the partition property $\omega_2\rightarrow[\omega_1]_{\aleph_1,<\aleph_1}^{<\omega}$. [2]
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 [4]):
- 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
- Donder, Hans-Dieter and Levinski, Jean-Pierre. Some principles related to Chang's conjecture. Annals of Pure and Applied Logic , 1989. www DOI bibtex
- Kanamori, Akihiro. The higher infinite. Second, Springer-Verlag, Berlin, 2009. (Large cardinals in set theory from their beginnings, Paperback reprint of the 2003 edition) www bibtex
- Jech, Thomas J. Set Theory. Third, Springer-Verlag, Berlin, 2003. (The third millennium edition, revised and expanded) www bibtex
- Eskrew, Monroe and Hayut, Yair. On the consistency of local and global versions of Chang's Conjecture. , 2016. arχiv bibtex