95 problems

Shelah's eventual categoricity conjecture: For every cardinal $\lambda$ there exists a cardinal $\mu(\lambda)$ such that if an AEC K with LS(K)${} \le \lambda$ is categorical in a…

Determine the automorphism group of the partial order of Turing degrees.

Let $T$ be a countable complete theory and let $P$ be a distinguished unary predicate in its vocabulary. Assume that $T$ fails the Gaifman property; that is, assume that there exis…

Let $L_{\text{or}}=(+,-,\cdot,<,0,1)$ be the language of ordered rings, and let $L_{\exp}=L_{\text{or}}\cup\{\exp\}$, where $\exp$ is a unary function symbol. Let…

Determine the least integer n for which no algorithm decides whether an arbitrary integer-coefficient polynomial in at most n variables has an integer zero. Source: William Gasarch…

Determine whether Con(ZFC plus the existence of a strongly compact cardinal) implies Con(ZFC plus the existence of a supercompact cardinal), or prove that the latter has strictly g…

Select arbitrary finite order k=1, rational constant coefficients a1,...,ak and rational initial values u0,...,u(k-1), each finitely encoded in binary, with u(n+k)=sum(i=1)^k ai u(…

Determine the exact large-cardinal consistency strength of ZFC plus the Proper Forcing Axiom (PFA): identify its equiconsistency level by matching upper and lower consistency bound…

Determine whether ZFR is consistent, where ZFR is the first-order theory ZF(j) plus the assertion that a nontrivial function symbol j:V-V is Sigma1-elementary and hence fully eleme…

Assuming delta is an extendible cardinal, prove that there is an inner model N contained in HOD such that N is a weak extender model for the supercompactness of delta and N satisfi…

What is the nature of the proof-theoretic ordinal (the smallest ordinal a theory cannot prove well-founded) for second-order arithmetic, ZFC, or stronger theories?

Is there a logic L which satisfies both the Beth property and Δ-interpolation, is compact but does not satisfy the interpolation property?

Is the theory of the field of Laurent series over $\mathbb{Z}_p$ decidable? of the field of polynomials over $\mathbb{C}$?

Is the Borel monadic theory of the real order (BMTO) decidable? Is the monadic theory of well-ordering (MTWO) consistently decidable?

If the class of atomic models of a complete first order theory is categorical in the $\aleph_n$, is it categorical in every cardinal?

Does a finitely presented homogeneous structure for a finite relational language have finitely many reducts?

Assume K is the class of models of a countable first order theory omitting countably many types. If K has a model of cardinality $\aleph_{\omega_1}$ does it have a model of cardina…

The universality spectrum problem: Is there a first-order theory whose universality spectrum is minimum?

The universality problem for C-free graphs: For which finite sets C of graphs does the class of C-free countable graphs have a universal member under strong embeddings?

Does every simple first-order theory have stable forking?

Shelah's categoricity conjecture for $L_{\omega_1,\omega}$: If a sentence is categorical above the Hanf number then it is categorical in all cardinals above the Hanf number.

The main gap conjecture, e.g. for uncountable first order theories, for AECs, and for $\aleph_1$-saturated models of a countable theory.

Over $\mathsf{RCA}^*_0$, is Ramsey's theorem for triples equivalent to $\mathsf{ACA}_0$, as it is over $\mathsf{RCA}_0$?

Determine the reverse-mathematical strength of Kříž's labeled-tree generalization of Kruskal's theorem.

Is Szemerédi's theorem provable in $\mathsf{ACA}_0$? More generally, determine its reverse-mathematical strength.