15 problems

Let $B_n$ be the Boolean algebra of all subsets of $[n]$, ordered by inclusion. A chain decomposition of $B_n$ has a type, namely the partition obtained by rearranging its chain si…

Let $i$ and $j$ be integers with $i\geq j\geq 2$, let $\mathbb{B}$ be a Boolean algebra, and let $n$ be the least integer greater than or equal to $i/(j-1)$. The quantities…

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.

For a theory $T$ and ordinals $\alpha\geq\beta$, say that a $\Pinf{\alpha}$ theory is non-splitting when it has exactly one $\Pinf{\alpha+1}$ theory as a descendant. In the case of…

Let $W$ be the Cantor set $\mathscr{D}_{1}$ and let $R=\operatorname{Co}(W)$ be the countable atomless Boolean algebra. Let $I$ be the set of finite binary strings, including the e…

Boolean-algebra decidability conjecture. The theory of Boolean algebras of cardinality less than $\lambda$, or the first-order theory of Boolean algebras expanded by quantification…

Let $\beta=\bigwedge_{n_0<\omega}\bigvee_{n_1<\omega}\cdots\omega^{k-1}\times A_{(n_0,n_1,\ldots,n_{k-1})}$ be the indicated iterated operation on a family of sets…

Let $\mathcal A$ be a Boolean algebra of sets, and say that it has the countable-chain (CC) property when every family of pairwise disjoint nonempty sets in $\mathcal A$ is countab…

Boolean-ultrapower saturation conjecture. There exist such $cappa$, $cB$, $U$, $cM$, and $cN$ for which

Let $\mathscr{A}$ be a Boolean algebra, and let $(CC)$ denote the condition referred to in the source. A Boolean algebra is a measure algebra if there exists a set function on it t…

Let $\mathcal B_N$ be the free Boolean algebra generated by $N\geq 1$ symbols $x_1,\dots,x_N$, with greatest element $X$ and least element $\emptyset$. Let $f\in\mathcal B_N$ be re…

A Boolean algebra $B$ is an algebra of subsets of a nonempty set, with the usual Boolean operations; write $B^+=B\setminus\{\mathbf 0\}$, and call a subset $A\subseteq B^+$ an anti…

Peckness conjecture. The edge poset of the quotient satisfies

Farah's PFA conjecture. The Proper Forcing Axiom, PFA, implies that all isomorphisms between any two quotient algebras of the form

The depth inequality conjecture. In ZFC,