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?