11 problems

Assume that $V$ satisfies $\mathsf{ZF}$ and that $\delta$ is an extendible cardinal. Let $G$ be $V$-generic for a forcing that collapses $V_\delta$ to be countable. Axiom of Choice…

Consistency conjecture. It is consistent with ZF plus the Axiom of Dependent Choices (DC) that BPI holds and there is no Vitali set in the real line.

In ZF, let a cardinal number be defined as an equivalence class under equinumerosity. For a graph, its chromatic index is the least cardinal number of colours in a proper edge-colo…

Work over $\operatorname{ZFU}$. Here SVC is the principle that there exists a set $S$ such that for every set $x$ there is an ordinal $\alpha$ with $x\preceq^* S\times\alpha$, and…

Let $AC_*^{1,1}$ and $AC^{1,1}$ denote the indicated choice principles in Henkin predicate logic, and write “weaker” for implication in HPL: a principle $P$ is weaker than $Q$ if e…

In set theory without assuming the Axiom of Choice, consider the various notions of well-foundedness for cardinals discussed in the paper. Well-foundedness-to-choice conjecture. At…

Let $M$ be a model of set theory satisfying the Axiom of Choice, and let $M'=M/\mathcal{F}$ be a permutation model inside $M$. Suppose that Frucht's Theorem fails in $M'$. An inter…

Let $(U,\mathcal S,\mathcal C)$ be a complete Mammen space, meaning that $U$ is a non-empty set, $\mathcal S$ is a perfect Hausdorff topology on $U$,…

Let $\phi_{IP}$ denote the choice principle asserting the existence of integer parts for real closed fields. Let $AC$ denote the axiom of choice. Non-implication conjecture. The pr…

In $\mathsf{ZF}$, an antichain of cardinals is a set of cardinals no two of which are comparable under the cardinal ordering. Feldman–Orhon conjecture. Every antichain of cardinals…

In the first-order language of set theory with equality and membership relation $\in$, count each individual quantifier rather than only quantifier alternations. Friedman's conject…