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4 Clarification, expanding the question

Suppose we are working in

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

• Is notion of wellfounded set absolute?
• Is notion of ordinal absolute?
• Is $\omega$ absolute?
• By absoluteness I mean absoluteness between transitive models of Consider the set theory defined above.

Those following notionsare 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?

By wellfounded sets I mean:

• Wellfounded sets

By ordinals I mean:

• $\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? 3 Clarification Suppose we are working in set theory without powerset, choice, and foundation. • Is notion of wellfounded set absolute? • Is notion of ordinal absolute? • Is$\omega$absolute? By absoluteness I mean absoluteness between transitive models of the set theory defined above. 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? By wellfounded sets I mean: $$WF(c) \Leftrightarrow (\forall x \subseteq TC(c)) \left[x \neq \emptyset \rightarrow (\exists y \in x) (\forall t \in x) [t \notin y]\right]$$ By ordinals I mean: $$ON(c) \Leftrightarrow WF(c) \wedge \text{Transitive}(c) \wedge (\forall x,y \in c)[x = y \vee x \in y \vee y \in x]$$ 2 Clarification Suppose we are working in set theory without powerset, choice, and foundation. • Is notion of wellfounded set absolute? • Is notion of ordinal absolute? • Is$\omega$absolute? 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?

By wellfounded sets I mean: $$WF(c) \Leftrightarrow (\forall x \subseteq TC(c)) \left[x \neq \emptyset \rightarrow (\exists y \in x) (\forall t \in x) [t \notin y]\right]$$

By ordinals I mean: $$ON(c) \Leftrightarrow WF(c) \wedge \text{Transitive}(c) \wedge (\forall x,y \in c)[x = y \vee x \in y \vee y \in x]$$

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