Is $\omega$ absolute in set theory without foundation? 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?
 A: Without the foundation axiom, you have to specify what you mean by an ordinal, precisely because the various definitions are no longer equivalent:


*

*An ordinal is a transitive set well-ordered by $\in$.

*An ordinal is a hereditarily transitive set. 

*An ordinal is a transitive set linearly ordered by $\in$. 


For example, in a model of Aczel's AFA, there is a set $a$ which is equal to $\{a\}$, and such a set is hereditarily transitive, but it is not well-ordered by $\in$, since it has no $\in$-least member, and it is not a strict linear order, since it is reflexive. One may similarly construct sets in AFA that are transitive and linearly ordered by $\in$, but not well-ordered. 
Similarly, the various equivalent formulations of well-foundedness become inequivalent without the axiom of choice. For example, the equivalence of the following two notions of well-foundedness is itself equivalent to the principle of dependent choice, a weak form of the axiom of choice:


*

*There is no infinite descending sequence

*Every subset has a minimal element. 


Thus, without any AC, the notion of well-foundedness depends on how you express it.
Another ambiguity here is that the meaning of ZFC-powerset is ambiguous without elaboration, as Victoria Gitman, Thomas Johnstone and I proved in our paper What is the theory ZFC-powerset?. Also, the familiar equivalent formulations of the axiom of choice (such as WOP or choice functions, etc.) are no longer equivalent when power set is absent. 
So it isn't clear exactly what your weak theory is.
A: 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:


*

*Maurice Boffa, Forcing et négation de l’axiome de Fondement, Académie Royale de Belgique, Mémoires, Classe des Sciences, Collection $8^o$, II. Série 40, No. 7 (1972).

*David Ballard and Karel Hrbáček, Standard Foundations for Nonstandard Analysis, Journal of Symbolic Logic 57 (1992), No. 2, 741–748.
