Mathematics Problem Archive

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

Is the complement of the finite spectrum of every first-order sentence also a finite spectrum? Equivalently, is $\mathrm{NE}=\mathrm{coNE}$?...

Does there exist a reasonable logic strictly stronger than first-order logic that has both compactness and Craig's interpolation property?...

Prove a superpolynomial lower bound on the size of Frege proofs; in particular, do some tautologies require exponentially large Frege proofs?...

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

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

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...

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...

Does every simple first-order theory have stable forking?...

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 ...

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

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

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

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

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

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

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

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