To be explicit, by Zermelo set theory with Choice, ZC, I mean the theory with the same language and axioms as ZFC except not Foundation (also called Regularity) and with the axiom scheme of Separation instead of Replacement. By Global Choice I mean adding a new function symbol $F$ and an axiom:$\forall v[v\neq\emptyset \rightarrow F(v)\in v]$, and extending the Separation scheme to include formulas using $F$.

It is known that the axiom of Global Choice gives a conservative extension of Zermelo Frankel set theory with Choice (ZFC). The proof I know is by Haim Gaifman in his "Local and Global Choice Functions" (Israel J. of Math v. 22 nos. 3-4, 1975, pp. 257--265. And there is one I have not worked through which uses forcing by Ulrich Felgner "Comparison of the axioms of local and universal choice" Fund. Math. 71, 1971, pp. 43-62. Both seem to require the idea of the rank of a set. Maybe one can be adapted to work for ZC, but I do not see it.

Or is there some other proof? Or a disproof?

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    $\begingroup$ Do you mean ZF+Global Choice, since you are referring to Replacement at the end of your first paragraph. Or do you mean Z+Global Choice, and Separation can refer to the choice function $F$? $\endgroup$ – David Roberts Mar 19 at 6:57
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    $\begingroup$ @DavidRoberts Sorry, that "replacement" was a typo for "separation." Corrected. $\endgroup$ – Colin McLarty Mar 19 at 11:26
  • $\begingroup$ thanks, that makes more sense. $\endgroup$ – David Roberts Mar 19 at 11:34
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    $\begingroup$ The proof by Felgner is not hard: take a model of ZFC and define a (proper class) forcing consisting of 'all partial well-orderings of the universe'. This forcing will add no new sets, but (by genericity) G will be a global well-ordering. Furthermore, Replacement with respect to G holds due to the Forcing Theorem (which holds for this particular class forcing) $\endgroup$ – Johannes Schürz Mar 20 at 15:19
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    $\begingroup$ @JohannesSchürz Yes, I believe Gaifman's proof is essentially the same. He just uses the fact that ZFC proves existence of enough partial well-orderings of the universe, that you do not really need forcing. $\endgroup$ – Colin McLarty Mar 20 at 15:52

The known proofs of conservativity of ZF + GC (Global Choice) over ZFC make significant use of replacement, and as far as I know the problem of conservativity of Z + GC over ZC is wide open.

Let me add that I have discussed the problem with a number of experts over the past two decades, and also posed it on FOM in 2006 in this posting.

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    $\begingroup$ This persuades me but I will leave the question open a little longer to see if it produces any surprises. $\endgroup$ – Colin McLarty Mar 20 at 15:53

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