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 ?



Are Z and ZC equiconsistent?
The **addendum** to my answer to [this related MO question](https://mathoverflow.net/questions/185338/is-there-an-l-like-inner-model-for-sf-z/185362) 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.


