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Let Z be Zermelo's system without choice (C) and without axiom of foundation (AF), namely extensionnality, pair, union, power set, separation, empty set, infinity. My question is the following one :

Is Z equiconsistent with Z+C ?

[I would also be interested it knowing whether Z+AF is equiconsistent with Z+AF+C, but I guess this is fairly close, and not as important to me]

I know that ZF is equiconsistent with ZF+C, so this is really a question without the replacement axiom.

I asked several specialists in set theory and they could not help me on this.

Any idea/reference on this question ?

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The addendum to my answer to this related MO question has a pointer to the work of Mathias (as pointed out by Avshalom's comment to my answer there) on a positive answer to the question, i.e., on establishing Con(Z + AC) assuming Con(Z) only.

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  • $\begingroup$ Thank you very much ! If I understand correctly, this provides a "yes" answer to both of my questions, right ? (I mean, with or without AF) $\endgroup$
    – Ivan Marin
    Commented Mar 31, 2023 at 18:51
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    $\begingroup$ Yes, I followed the modern practice of including AF among the axioms of Z, although Zermelo's original formulation allowed urelements. $\endgroup$
    – Ali Enayat
    Commented Mar 31, 2023 at 21:34

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