DMC versus DC over ZF remains open

Over $\mathrm{Z}\mathrm{F}$: DC implies DMC. Strictness is known in $\mathrm{Z}\mathrm{F}\mathrm{A}$, but whether DMC implies DC in $\mathrm{Z}\mathrm{F}$ is open; no strictness over $\mathrm{Z}\mathrm{F}$ is asserted.

• The implication. DC implies DMC over $\mathrm{Z}\mathrm{F}$ by DC and finite multiple selections, which records the equivalence $\mathrm{D}\mathrm{C}\leftrightarrow (\mathrm{D}\mathrm{M}\mathrm{C}\wedge {\mathrm{A}\mathrm{C}}_{\omega ,\mathrm{f}\mathrm{i}\mathrm{n}})$ and hence gives the direction used here. The principles are those of The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain and Dependent multiple choice in finite-level tree form; the finite-selection form is the one in which the implication is stated, so no conversion between the tree and menu presentations is needed.

• The $\mathrm{Z}\mathrm{F}\mathrm{A}$ strictness. Dodu and Morillon record that Fraenkel's second model of $\mathrm{Z}\mathrm{F}\mathrm{A}$ satisfies DMC but does not satisfy DC. Thus the model has a DMC menu system for every serial relation while some serial relation has no infinite dependent-choice chain. This is the direction needed to show that DMC does not imply DC in $\mathrm{Z}\mathrm{F}\mathrm{A}$. It is not a statement about $\mathrm{Z}\mathrm{F}$, and no such statement is made here.

• The open question. Morillon poses the reversal over $\mathrm{Z}\mathrm{F}$ as an open question in the same section that records the tree form of DMC. This item reports that status and nothing more: absence of a published proof of $\mathrm{D}\mathrm{M}\mathrm{C}\Rightarrow \mathrm{D}\mathrm{C}$ in $\mathrm{Z}\mathrm{F}$ is not converted into a nonimplication theorem, and the separately proved independence results of this page, which refute Urysohn's lemma under principles that do not imply DMC, do not decide the reversal either.

• Why the qualification matters for consumers. The published Baire-category ledger on the deferred catalogue asserts strictness of DMC below DC and below multiple choice in $\mathrm{Z}\mathrm{F}$ and $\mathrm{Z}\mathrm{F}\mathrm{A}$ alike. That stronger wording is not a theorem here: the results proved on this page give the implication and the $\mathrm{Z}\mathrm{F}\mathrm{A}$ separation, and consumers of this item must keep the two theories apart.