Q:

How badly does foundation fail in NF(etc.)?

The strongest antifoundation axiom I know is due to Boffa. Roughly, it asserts that every graph which could represent a set, does. For example, considering a graph consisting of (a root connected to) arbitrarily many "one-vertex loops" leads to the realization that Boffa implies the existence of a proper class of distinct Quine atoms (= sets satisfying $x=\{x\}$); by contrast, Aczel's antifoundation axiom implies that there is exactly one Quine atom.
For details, see Aczel's book on the subject, especially chapter $5$.
Motivated by this question, I want to ask about the situation in Quine's set theory and its variants (for simplicity, I'll treat urelements as Quine atoms):

Given a model $M$ of NF or one of its variations (e.g. NFU), what can we say about $(i)$ the set $\mathcal{TRAN}(M)$ of directed graphs in $M$ which are isomorphic in $M$ to graphs of the form $(X, \in\upharpoonright X)$ for $X$ transitive? And $(ii)$ the set $\mathcal{SET}(M),$ where we drop the requirement that $X$ be transitive?

(Note that the consistency of NF, even relative to large cardinals, is currently open as far as I know, with Holmes' claimed proof not yet vetted; for the purposes of this question, though, I'm assuming it.)

A:

This is quite messy, actually.   NF doesn't prove the existence of many transitive sets, and the restriction of $\in$ to a transitive set is a set very rarely.  So your set TRAN might turn out to have very little in it.  It might not contain any infinite sets for example.   Specifically the restriction of $\in$ to the transitive set $V$ is provably not a set, so the graph of which $V$ is a picture literally doesn't exist.  I suspect the question you have at the back of your mind is subtly different.

