MathOverflow

C.S. Peirce presented an ideosyncratic and graph-based version of algebraic logic; the number $3$ shows up quite a bit, including in the "reduction thesis" that roughly says that binary relations are not enough to "build everything" but ternary relations...

For finite families of sets $\mathcal{F} = \{A_1, \ldots, A_n\}$ define a function:
$$e(\mathcal{F}) = \left| \left\{\{B, C\}: B, C \in \mathcal{F}, B\neq C, B\cup C \in \mathcal{F} \right\}\right|$$
$\mathcal{F}$ is union-closed if and only if $e(\mathcal{F}) = \binom{n}{2}$....

Background. A set $X$ is Dedekind-finite, or D-finite, if every injection $\iota:X\to X$ is surjective. In ${\sf ZF}$, if $X$ is a D-finite set, its powerset ${\cal P}(X)$ is not necessarily D-finite. Which prompts the following
Question. In ${\sf ZF}$, is it consistent that there is a D-finite set $X$ and a bijection $\varphi:{\cal P}(X)\to\omega$?...

Is there any formal precedent for identifying the maximum length of strictly ascending chains of ideals in a bounded Noetherian ring with the maximum depth of a finite-branching perfect-information game tree? I am investigating a framework where both...