7 problems

Let $\mathsf{MSO}^\phi(\sharp)$ be monadic second-order logic with the Härtig quantifier, and call an arithmetical predicate directly definable when it is defined by the logic in t…

Let $\operatorname{ULST}(\mathbb{L}^2)$ denote the upward Löwenheim–Skolem–Tarski number of full second-order logic, when it exists: the least cardinal $\kappa$ such that every suf…

The ULST conjecture. The strength of the existence of $\operatorname{ULST}(\mathbb L^2)$ is exactly that of an extendible cardinal; equivalently, the $\operatorname{ULST}$ number f…

Critical-period conjecture. The critical period of set theory will be realized with the advent of second-order logic and the revival of Hilbert's program.

Let $\mathcal{Y}$ be the second-order sentence formalizing Yablo's paradox, and let $\neg\mathcal{Y}$ denote its negation. A first-order theory is a set of first-order sentences in…

Internal normalization conjecture. A normalization argument for $\mathbf{ITT}_V$ might be formalizable in $\mathbf{ITT}_V$ itself.

Let $NDL$ be the generalized dynamic logic discussed in the paper, and let Henkin-style second-order logic (HSOL) be the corresponding Henkin semantics for second-order logic. A pr…