73 problems

A type-two well-ordering principle is a well-ordering principle whose transformation of linear orders is of type two, together with the assertion that the resulting order is well-f…

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…

Solovay's conjecture. ${\bf FIM}$ has the same consistency strength as ${\bf BI}$.

Let $c:\mathbf{N}\to 2$ be a coloring, and call an infinite set $H$ homogeneous for sums of length at most $n$ if all sums of between one and $n$ distinct elements of $H$ have the…

Wang's conjecture. Cohesiveness and the Erdős–Moser theorem each admit preservation of the arithmetic hierarchy.

A measurable set $P\subseteq 2^\omega$ is regular when there are a $G_\delta$ set $Q\supseteq P$ and an $F_\sigma$ set $S\subseteq P$ such that $\mu(S)=\mu(P)=\mu(Q)$. Here a…

Let $\textup{\textsf{open}}$, $\textup{\textsf{HBU}}$, $\mathbb{Z}_{2}^{\omega}$, and $\textup{\textsf{QF-AC}}^{0,1}$ denote the principles and systems used in the paper. Non-impli…

Let $X$ be a complete metric space, let $F\subseteq X$ be a nonempty closed set, and let $\{U_n\}_{n\in\omega}$ be a sequence of nonempty open sets in $X$ such that each $U_n$ is d…

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…

2-dimensional hyperimmunity conjecture. If a Ramsey-like theorem does not preserve one 2-dimensional hyperimmunity, then it implies

Let a set be fractally countable relative to a base system when it is obtained as a union of countable sets definable in a sequence of extensions of that system, as in … Let $HYP$…

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

Finite-$\Sigma^1_1$-choice non-derivability conjecture. The principle \textup{\textsf{finite-}\Sigma^1_1\text{-AC}_0} cannot be obtained from item (ii).

Finite Choice non-derivability conjecture. The principle $\textup{\textsf{Finite Choice}}$ cannot be obtained from the second item.

Reverse-mathematical conjecture. Over $\mathrm{RCA}_0$, the existence of an existentially closed Fraïssé class of groups is equivalent to $\mathrm{WKL}_0$, and the existence of an…

Let ${\textsf{\textup{Z}}}_{2}^{\omega}$ be the theory denoted by ${\textsf{\textup{Z}}}_{2}^{\omega}$, and let $\textup{\textsf{OC}}^{0,0}$ and $\textup{\textsf{ClC}}$ be the choi…

Let $n\in\omega$ be the number of applications of the hyperjump operator used to prove a theorem provable from $\Pi^1_1\mathchar`-\mathsf{CA}_0$. Hyperjump-iteration classification…

Conjecture. The method presented in the paper can be extended to relate the Ramsey theorem for bicolorings of exactly $lpha$-large sets to the systems $Pi^0_{\omega^\beta}$-…

In second-order arithmetic, let $\textsf{RCA}_0$ and $\Pi^1_1\textsf{-CA}_0$ denote the indicated subsystems, and consider the theorem that inverse Goodstein sequences terminate, w…

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 $\textup{\textsf{RCA}}_{0}^{\omega}$ be the higher-order base theory of recursive comprehension and let $\textup{\textsf{WWKL}}$ denote weak weak König's lemma. Let Theorem be…

Let $\mathsf{WKL}_0$ denote weak König's lemma and let $\mathsf{RCA}_0$ denote the usual base theory. WKL₀ preservation conjecture. Many basic theorems, including…

Let $\mathsf{ACA}_0$ be arithmetical comprehension and let $2^{\mathbb{N}}$-$\mathsf{ACA}_0$ denote the corresponding higher-type principle used in the paper. ACA₀ transfer conject…