43 problems

A finite effective procedure is an algorithm that terminates after finitely many steps and follows an effective, mechanically executable rule. Alan Turing's theoretical machines ar…

The paper discusses formal versions of epistemological antinomies and Tarski's theorem on the undefinability of truth. Epistemological-antinomies conjecture. The formal version of…

The article considers mathematics, logic, and related disciplines, together with the processes of multiple alignment, unification, and search. Information-compression conjecture. M…

Consider the Nelson–Oppen combination procedure for two component theories, where stable infiniteness is the standard requirement ensuring its correctness. Tinelli–Harandi conjectu…

Let a theory have the property introduced by Toledo, Zohar, and Barrett in 2023, which is described as seemingly weaker than strong politeness and sufficient for combination with a…

Let the internal language of derived $\Gamma$-geometry be the language associated with its $\Gamma$-topoi, and let dependent type theory be extended by a ternary connective modelin…

Consider the diagram of constructive quantum logics, whose nodes include classical logic, intuitionistic logic, orthologic, Ex-logic, fundamental logic and the intermediate logics…

Turing's conjecture. For every arithmetical theorem $\varphi$, there is a computable presentation of an ordinal $\alpha$ such that

Gödel–Hilbert programme conjecture. Gödel may have intended to argue that, for every system of the specified kind, the sentence $G$ is axiomatically undecidable but is decidable vi…

Number-sort consequence conjecture. It is conjectured that $\mathsf{V}^1_2$ has more number-sort consequences than all the other theories mentioned so far.

Consider theories such as ZFC and an isomorphism between sentences that are impossible to prove and families of sentences that are hard to prove efficiently. True-unprovable-senten…

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…

Decidability conjecture. All statements about physical observables in tame quantum field theories are decidable.

The paper considers the preceding question, whose notation and precise formulation are not included in the supplied passage. Positive-answer conjecture. The question has a positive…

Pearl–Paz conjecture. There exists a finite set of axioms characterizing all valid conditional independence implication statements.

Let $Q$ be a propositional proof system. A theory $\mathcal{T}$ is a conservative extension of a base theory when it extends that theory without proving any new sentences in the ba…

Cieśliński–Urbaniak's conjecture. Any two distinct instances $Y^R(\overline{m})$ and $Y^R(\overline{n})$ are not provably equivalent.

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 $\langle X,\tau,\mathbf{B}\rangle$ be an $\mathcal{S}$-Priestley space. For subsets $U,V\subseteq X$, write $V^{c}=X\setminus V$ and let $\downarrow W$ denote the down-set gene…

Let $\langle X,\tau,\mathbf{B}\rangle$ be an $\mathcal{S}$-Priestley space, and let $X_{B}$ denote the subset associated with $\mathbf{B}$ in the Priestley duality. The clopen up-s…

The paper considers programs that notate increasingly large ordinals, including constructions reaching beyond previously developed techniques. Intelligence-growth conjecture. Each…

Let $L$ be a language and let $S(x)$ be an $L$ clause set. Refutability by a clause set cycle is the property that $S(x)$ has a clause set cycle refutation, while open induction is…

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…