Mathematics Problem Archive

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?...

For any finite sets $X$ and $Y$ of published mathematical theorems expressible in second-order arithmetic, is either $\mathsf{RCA}_0+X$ interpretable ...

Are the following statements equivalent to $\Pi^1_1\text{-}\mathsf{CA}_0$? (i) If countable torsion abelian groups $G,H$ satisfy $G\oplus G\cong H\opl...

Determine the reverse-mathematical strength of Alexandroff's one-point compactification theorem for countably based MF spaces....

Determine the reverse-mathematical strength of the assertion that a proper MF space is metrizable if and only if it is regular....

Over $\mathsf{RCA}_0$, does the Furstenberg–Zimmer structure theorem imply $\Pi^1_1\text{-}\mathsf{CA}_0$?...

For an ordinal $\alpha$, determine the length (maximal order type) of the well-quasi-order $L_\alpha$ of countable linear orders of Hausdorff rank bel...

Determine the reverse-mathematical strengths of Laver's labeled-linear-order theorem $\mathsf{LAV}$ and the Nash–Williams bqo transfinite-sequence the...

Is weak-$L_{\omega_1,\omega}$-$\mathsf{CA}$ equivalent to weak-$\Sigma^1_1$-$\mathsf{AC}_0$?...

Is the open mapping theorem for separable Banach spaces provable in $\mathsf{RCA}_0$, or at least in $\mathsf{WKL}_0$?...

Determine the exact reverse-mathematical strength of the Krein–Šmulian theorem for separable Banach spaces....

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

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

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

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

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...