16 problems

Pakhomov–Freund conjectures. The following two equivalences are conjectured:

Let $\operatorname{KT}_{\ell}(\omega)$ denote labelled Kruskal's theorem with labels drawn from $\omega$, and let $\operatorname{KT}_{\ell}(n)$ denote its restriction to labels bel…

$\Sigma^1_2$-altitude characterization conjecture. $\mathsf{Alt}_{\Sigma^1_2}(R)$ is equal to the least height of such a transitive model $M$.

Let $\mathsf{COH}$ denote the cohesive principle, let $\operatorname{\mathsf{Ind}}\mathscr{E}_k$ denote the indicated indivisibility problem for $k$-colorings, and let…

Bounded-arithmetic conjecture for Arrow's theorem. The first-order formalisation of Arrow's theorem is provable in

Let $\textup{\textsf{NIN}}$ denote the problem of finding a function witnessing non-injectivity, and let $\textup{\textsf{HBU}}$ denote the Heine–Borel theorem for covers of…

Let $T$ be a strong system with an ordinal analysis based on ordinal representation systems $\mathsf{OT}$. These systems give rise to functors … that send dilators to ordinal repre…

Finite-path perfect matching conjecture. $ext{Finite Path }\mathsf{PM}$ is equivalent to $peratorname{\mathsf{ATR}}_0$ over $peratorname{\mathsf{RCA}}_0$.

The computational-strength conjecture. $athsf{MM}$ and $athsf{MIM}$ are equivalent over $peratorname{\mathsf{RCA}}_0$, i^1_1\text{-}\mathsf{TR}}_0 proves $athsf{PM}$, and…

Nonuniform computability conjecture. There is an $M_{0}^{3}$ satisfying $\textup{\textsf{PR}}(M_{0})$ such that no $\Theta^{3}$ satisfying $\textup{\textsf{SCF}}(\Theta)$ is comput…

Giusto–Simpson conjecture. The following are equivalent over $\mathsf{RCA}_0$:

Let $A$ be a set of natural numbers, let $A'$ denote its Turing jump, and let $0”$ denote the second Turing jump of the computable degree. A set is almost everywhere dominating whe…

Preservation conjecture. Both $coperatorname{COH}$ and $coperatorname{EM}$ admit preservation of the arithmetic hierarchy.

Gasarch–Hirst conjecture.

Let $T$ be one of … An ultrafilter $\mathfrak{U}$ is a $P$-point if, for every partition $\mathbb{N}=S_0\cup S_1\cup\cdots$ with $S_n\notin\mathfrak{U}$ for every $n$, there is an…

An IP set is a set of natural numbers containing all finite sums of an infinite sequence of natural numbers. Let $\mathfrak{U}$ be an ultrafilter on $\mathbb{N}$, and let…