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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, 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, Journal of Symbolic Logic 57 (1992), No. 2, 741–748.
show/hide this revision's text 1

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., $\omega^M\ne\omega$.