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The "conservative" class theory, NBG, proves no new theorems about sets (with respect to ZFC). The choice function used here is set choice, and it's not too hard to prove (if M is a ctm for ZFC, then D(M) is a ctm for NBG and has the same set universe).

However, if we add the axiom that there is a bijection (in the class universe) between $\mathbf{V}$ and $\mathbf{ON}$, the classes of sets and ordinals, respectively, this is apparently still conservative over ZFC (with a much stronger class choice axiom). However, I can't find a reference for this.

Apparently this fact is credited to Easton and Solovay, published by Easton in 1964, and apparently it uses class forcing, but I can't find any more specific information on this topic, or the paper itself. Does anyone have more specific information on this, or better search skills than I do?

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    $\begingroup$ I don't have the reference handy, but the forcing consists of all set-sized partial injections from $\mathbf{ON}$ to $\mathbf{V}$, ordered by extension. Since this is $\kappa$-closed for all $\kappa$, no new sets are added in the forcing extension, and the generic is a bijection from $\mathbf{ON}$ to $\mathbf{V}$. $\endgroup$
    – François G. Dorais
    Commented Apr 6, 2012 at 15:50
  • $\begingroup$ This is the forcing I wanted to do, but unfortunately I lacked the background to prove (to my own satisfaction) that it actually works. $\endgroup$
    – Richard Rast
    Commented Apr 6, 2012 at 16:12
  • $\begingroup$ See also Victoria Gitman's nice blog post containing a full account of this proof: boolesrings.org/victoriagitman/2013/10/09/… (and click through to the second post) $\endgroup$
    – Joel David Hamkins
    Commented Feb 19, 2017 at 11:51

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Yes, NBG + Equipollence, which is equivalent to NBG + Global Choice, is a conservative extension of ZFC; this was independently discovered by many people (at least Cohen, Felgner, Grishin, Jensen, Kripke, and Solovay). A detailed proof can be found in

Ulrich Felgner: Comparison of the axioms of local and universal choice, Fundamenta Mathematicae 71 (1971), pp. 43–62 (EUDML, PLDML).

The basic idea of the proof is pretty straightforward: you take the class of injective (set) functions whose domain is an initial segment of On as your forcing conditions, and then a generic filter gives you a bijection of On and V. (Actually, Felgner uses choice functions as forcing conditions, but that does not make much of a difference.)

The result is also included in the follow-up surveyish paper

Ulrich Felgner: Choice functions on sets and classes, in Sets and classes: On the work of Paul Bernays (Gert H. Müller, ed.), North-Holland, Amsterdam, 1976, pp. 217–255, doi 10.1016/S0049-237X(08)70895-4.

This paper also discusses a generalization of the result to set theory without the foundation axiom ($\mathrm{ZFC}_-$, $\mathrm{NBG}_-$). It turns out that in absence of foundation, equipollence has nontrivial consequences even for sets: $\mathrm{NBG}_- + \mathrm{Equipollence}$ is a conservative extension of $\mathrm{ZF}_-$ extended by the schema $$\forall x\:\exists y\:\phi(x,y)\to\forall\alpha\in\mathrm{On}\:\exists f\colon\alpha\to\mathrm{V}\:\forall\beta<\alpha\:\phi(f\restriction\beta,f(\beta))$$ (a sort of class version of $\mathrm{DC}_\alpha$; an equivalent formulation: any class tree whose height is bounded by an ordinal has a maximal path). This schema implies AC and collection (assuming replacement), but one can show that it is strictly stronger than that (i.e., independent of $\mathrm{ZFC}_- + \mathrm{Collection}$).

Note that without foundation, equipollence still implies global choice, but not vice versa. It is apparently an open problem if $\mathrm{NBG}_- + \mathrm{Global\ Choice}$ is conservative over $\mathrm{ZFC}_-$.

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