Let $\text{ZF}^-$ be the set theory without powerset, choice, and foundation. Consider the following notions:


* Wellfounded sets
$$WF(c) \Leftrightarrow (\forall x \subseteq TC(c)) \left[x \neq \emptyset \rightarrow (\exists y \in x) (\forall t \in x) [t \notin y]\right]$$
* Ordinals
$$ON(c) \Leftrightarrow WF(c) \wedge \text{Transitive}(c) \wedge (\forall x,y \in c)[x = y \vee x \in y \vee y \in x]$$
* $\omega$
$$x = \omega \Leftrightarrow ON(x) \wedge (\forall y \in x)[y = \emptyset \vee (\exists z)[y = z \cup \{z\}]] $$


Are those $\Delta\_1$ notions? Are those notions absolute between transitive models of $\text{ZF}^-$? More precisely, is there a model $V$ of $\text{ZF}^-$ with transitive classes $N,M \models \text{ZF}^-$ s.t. those notions are not absolute between $N$ and $M$? Does the situation change if we add powerset or choice to our theory?


Those notions are absolute once we have foundation, moreover they are definable by a $\Delta\_0$ formula. I was wondering if they are still absolute without assuming foundation, but that requires more work to show; or if they cease being absolute at all. If so, is there an easy way to construct a counterexample?



Is $\omega$ absolute in set theory without foundation?
None of these three notions is absolute, even if you retain powerset and the axiom of choice.


In Boffa’s set theory (which contains ZFC without foundation, and is conservative over ZFC with respect to the well-founded kernel), every extensional set-like binary relation is isomorphic to a transitive class with $\in$. In particular, you can take the ultrapower of the universe over a nonprincipal ultrafilter on a countable set, and let $M$ be its transitive collapse. Then there are nonstandard integers in $M$, i.e., $\omega^M\ne\omega$.


References for Boffa’s axiom:


1. Maurice Boffa, [*Forcing et négation de l’axiome de Fondement*](http://www.ams.org/mathscinet-getitem?mr=476514), Académie Royale de Belgique, Mémoires, Classe des Sciences, Collection $8^o$, II. Série 40, No. 7 (1972).
2. David Ballard and Karel Hrbáček, [*Standard Foundations for Nonstandard Analysis*](http://www.jstor.org/stable/2275304), Journal of Symbolic Logic 57 (1992), No. 2, 741–748.


