18 problems

Presburger Arithmetic, denoted by $\mathbf{PrA}$, is the true theory of natural numbers with addition. A theory is reflexive when it can prove the consistency of all its finitely a…

Pretransitivity conjecture. If a normal modal logic $\Lambda$ is interpretable by a modal-to-modal translation with parameters in a pretransitive normal logic, then $\Lambda$ is pr…

Non-interpretability conjecture. The structure $(\mathbb{Z};+,C_\alpha)$ does not interpret $(\mathbb{Z};+,C_\beta)$, and conversely.

Let $\mathsf{WSeq}$ be the weak sequence theory and let $\mathsf{R}$ be Robinson's arithmetic. Mutual interpretability conjecture. The theories $\mathsf{WSeq}$ and $\mathsf{R}$ are…

Let $\mathsf{T}$ be the tree theory and let $\mathsf{Q}$ be Robinson arithmetic. Kristiansen–Murvanashyaka conjecture. The theory $\mathsf{Q}$ interprets $\mathsf{T}$. They prove t…

Let von Neumann's function theory be the formal theory referred to in the surrounding discussion, and let $\mathrm{NBG}$ denote von Neumann–Bernays–Gödel set theory. Two theories a…

Minimal evidentness conjecture. For each interpretability degree, there may be statements that are minimal in the evidentness order.

Forward-map memorization conjecture. Descrambling a network using purely noisy data still permits an interpretation of its weights because the forward map is memorized in the right…

Irrational cyclic-order conjecture. The structure $(a0;+,C_{a0})$ does not interpret $(a0;+,C_{b2})$, and conversely.

Let $\sf G1$ denote Gödel's first incompleteness property, and order recursively enumerable theories by interpretability, written $\lhd$. A theory is minimal for $\sf G1$ when it s…

Let $A$ be a finitely axiomatized sequential theory and let $N_0:{\sf S}^1_2\lhd A$. For an interpretation $M:A\rhd A$, let $\rho(M)$ denote its complexity, and let $\mathcal I_n$…

Let $A$ be a finitely axiomatized Vaught theory in signature $\Theta_0$, let $N:A\rhd{\sf R}$, and suppose $\top\rhd_A{\sf TB}^{-}_N$. Finite-approximation conjecture. There is a…

Let $U$ be a Vaught theory and let $N:{\sf R}\lhd U$. For theories or axiom sets $\alpha$ and $\Gamma$, write \alpha\mathrel{\text{\textcolor{gray}{blacktriangleright…

Let $A$ be finitely axiomatized, consistent, and sequential, and let $N:{\sf S}^1_2\lhd A$. Write $A+{\sf TB}^{-}_N$ for the corresponding Tarski-biconditional extension. Mutual-in…

Let $U$ be a Vaught theory, meaning a theory with the relevant direct interpretation properties described in the paper. Call $U$ Enayat when it interprets itself together with the…

Let $U$ be a sequential theory. It is Enayat when it interprets itself together with the Tarski biconditionals for its language. Sequential Enayat-theory conjecture. There are no c…

Minimal interpretation-degree conjecture. There is no theory with a minimal degree of interpretation for which $\sf G1$ holds, $\langle \sf D, \lhd\rangle$ is not well founded, and…

Non-locally modular interpretation conjecture. The theory $T$ interprets an algebraically closed field.