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

Question

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.)

Answer 1

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.

Answer 2

If you are interested in how badly foundation fails in NF(U), try looking at it like this. We know that $V \in V$, so foundation fails. That is: there are classes (sets, even) that lack $\in$-minimal members. One might ask: is $V$ the only reason for the existence of such collections? Might it be the case that every (``bottomless'') set lacking an $\in$-minimal member contains $V$?. Such a hypothesis would exclude Quine atoms, and we know how to exclude them anyway. At this stage it seems possible that we could find models of NF in which every bottomless class contains $V$ - so that the existence of $V$ is the only thing that falsifies Foundation.

Is this the kind of thing you are after?