I'm currently thinking about how different models of set theory view each other. In particular I'm looking at how well-foundedness behaves between different models.
So we have the Axiom of Regularity:
$\forall x [x \not = \emptyset \rightarrow \exists y \in x (y \cap x = \emptyset)]$
We know also (by your choice of Compactness for first-order theories, Mostowski Collapse, or Ultrapower construction), that this axiom (and indeed the whole of $ZFC$) can be satisfied where the particular relation being interpreted as membership is not, strictly speaking, well-founded (at least from the perspective of what we normally take to be standard models).
I'm intruiged by a principle of Joel Hamkins' Multiverse View (given in [Hamkins 2012]):
Well-foundedness Mirage. Every universe $V$ is non-well-founded from the perspective of another universe.
I'm just trying to get my hands on a model of $ZFC$ that thinks that `a standard model' (if I may be excused the absolutism for a minute) of $ZFC$ is non-well-founded. Thus far I just can't quite crack it. A compactness construction would seem to be easiest, but I'm struggling to generate one that thinks the standard model is non-well-founded. Any hints/references/examples would be really gratefully received!
References: [Hamkins 2012] Hamkins, Joel-David, `The Set-Theoretic Multiverse', THE REVIEW OF SYMBOLIC LOGIC, Volume 5, Number 3, September 2012