Recently I saw the abstract of a paper by Nash-Williams: ``**Should axiomatic set theory be translated into graph theory?**''. The abstract, taken from Mathscinet says the following:

The author translates the Zermelo-Fraenkel axioms for set theory ($ZF$) into graph-theoretic language by replacing "set'' by "vertex'' and the membership relation "$∈$'' by the preneighborhood relation "$→$''. A digraph $D$ is an ordered pair $(V(D),E(D))$ such that $V(D)$ is a class (of vertices) and $E(D)⊆V(D)×V(D)$ is the subclass of ordered pairs $(ξ,η)$ such that $ξ$ is a preneighbor of $η$. He describes those graph-theoretic properties a digraph must satisfy in order to be a model of $ZF$ (with or without the axiom of choice ($AC$)). The theorem that $AC$ is independent of $ZF$ can then be formulated as a theorem of graph theory asserting the existence of certain kinds of digraphs.

The author proposes that a translation of this and other independence proofs might yield further insight into why the theorems are true. He also suggests that a textbook on axiomatic set theory as the study of digraphs satisfying certain conditions might increase its accessibility to nonspecialists. However he cautions that translations of independence theorems from one language to another must justify their existence by achieving some substantial progress in either set theory or graph theory.

Now my question is:

**Question.** Are there any extra works regarding the authors's suggestion ``other independence proofs might yield further insight into why the theorems are true.''?