10 problems

Shelah's three cardinal and model-existence claims. The following assertions hold:

Characterizability conjecture.

Let $m\geq 1$. Write $\mathcal{A}_{m+1}$ for the canonical tree of rank $m+1$, and let $B^{(m)}_j:=[m+1:j]$ be the tree obtained by deleting all but $j$ of the level-one subtrees o…

Let $\psi$ be a sentence of $L_{\lambda^+,\omega}$ in a language of size $\lambda$. Categoricity means having exactly one model, up to isomorphism, in a given cardinality. Shelah's…

Let $\mathrm{L}$ be a relational $\omega$-signature. A theory in $\mathrm{L}_{\infty\infty}$ is Boolean satisfiable if it has a Boolean-valued model satisfying the theory. A Boolea…

Eventual categoricity conjecture. There is a threshold cardinal $ctheta$ such that if $cpsi$ is categorical in some $cmu cge ctheta$, then $cpsi$ is categorical in all…

Let $K$ be a class of finite structures, let $t(n)$ be polynomially bounded, and let $r(s, \mathcal{A})$ denote the relevant iteration-depth parameter for a structure $\mathcal{A}$ a…

Let $\phi$ be an $\mathcal{L}_{\omega_{1},\omega}$-sentence of quantifier depth $\alpha$, and let Scott rank refer to the Scott rank of a model of $\phi$. Uniform Vaught's conjecture. T…

Cofinality conjecture. For $\kappa$ a characterizable cardinal, $\kappa$ is not homogeneously characterizable if and only if

Shelah's conjecture. Every ${\mathcal{L}_{\omega_1,\omega}}$-sentence with a model of cardinality $\aleph_{\omega_1}$ has a model of cardinality $2^{\aleph_0}$.