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show/hide this revision's text 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?
    show/hide this revision's text 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: $$ON(c) \Leftrightarrow WF(c) \wedge \text{Transitive}(c) \wedge (\forall x,y \in c)[x = y \vee x \in y \vee y \in x]$$
    show/hide this revision's text 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: $$ON(c) \Leftrightarrow WF(c) \wedge \text{Transitive}(c) \wedge (\forall x,y \in c)[x = y \vee x \in y \vee y \in x]$$
    show/hide this revision's text 1