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NBG is a conservative extension of ZFC that includes a concept of "proper class." Now I like the conservativity, since it means anytime I want to prove something in ZFC, I am free to work in NBG. However, the NBG approach to proper classes isn't very "malleable;" e.g. we cannot have one proper class as an element of another proper class, we cannot exponentiate proper classes etc.

So what I'd like is a (material) set theory with proper classes that can be manipulated as if they were sets, which is nonetheless conservative over the language of ZFC. Is this possible? Somehow we have to block the "consistency theorem" for proper class sized models; if $V$ is the proper class of all sets and $\in$ is the membership relation on $V$, then despite that $(V,\in)$ is a model of ZFC, nonetheless we don't want to deduce that ZFC is consistent, since this is a proper class model.

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Yes, you can have a full ZFC-like hierarchy on top of the universe, with no consistency-strength penalty.

Consider the theory denoted ZFC + $V_\kappa\prec V$, expressed in the language with an additional constant symbol $\kappa$, where $V_\kappa\prec V$ denotes the scheme $$\forall x\in V_\kappa\left[\varphi(x)\iff \varphi^{V_\kappa}(x)\right].$$ Thus, the scheme $V_\kappa\prec V$ expresses that $\kappa$ is a correct cardinal.

This theory lines up with your theory, if you think of the elements of $V_\kappa$ as the real "sets" and the elements of rank $\kappa$ and higher as the classes and meta-classes and so on.

Theorem. The theory ZFC+$V_\kappa\prec V$ is equiconsistent with ZFC. Indeed, every model $M\models\text{ZFC}$ has an elementary embedding $M\prec V_\kappa^{\bar M}$ into a model $\bar M$ of the theory, so that $V_\kappa^{\bar M}\prec \bar M$.

Proof. This is a simple compactness argument, applied with the reflection theorem. Let $T$ be the theory that augments the elementary diagram of $M$ with the assertions $a\in V_\kappa$ for each $a\in M$ and $\forall x\in V_\kappa[\varphi(x)\iff\varphi^{V_\kappa}(x)]$. It follows by reflection that every finite subtheory of $T$ is realized inside $M$, and so $T$ is consistent. Any model of $T$ provides the desired $\bar M$. QED

Thus, the theory is conservative over ZFC, and proves no new facts about "sets", since anything true in any model $M$ of ZFC is also true in the sets, that is in $V_\kappa$, of a model of ZFC+$V_\kappa\prec V$.

By essentially the same argument, one can get a transfinite tower of such elementary substructures $V_\kappa\prec V_\lambda\prec V$ for all $\kappa,\lambda\in C$, where $C$ is a closed unbounded class of cardinals. This is known as the Feferman theory.

This idea has appeared here on MO a few times, such as in my answer concerning various weak universe concepts, an answer by François Dorais, another, and (probably the most relevant) my answer about elementary substructures of the universe.

Note one subtle issue of the theory here, though, is that $\kappa$ might be singular. To insist that $\kappa$ is regular will imply that it is inaccessible, and this then will no longer be a conservative extension. If you want $V_\kappa\prec V$ where $\kappa$ is inaccessible, then this is called a reflecting cardinal, which is equiconsistent with and conservative over the theory known as Ord is Mahlo. Every model of Ord is Mahlo elementarily embeds into the rank initial segment $V_\kappa$ of a reflecting cardinal $\kappa$ inside a model of $V_\kappa\prec V+\kappa$ inaccessible.

Another subtle point addresses your concern at the end. Namely, even though $V_\kappa\prec V\models\text{ZFC}$, you cannot deduce from this that $V$ knows $\text{Con}(\text{ZFC})$, since you only have $V_\kappa\prec V$ as a scheme. You don't have any statement from which you can conclude in $V$ that $V_\kappa$ satisfies all the ZFC axioms, only that it satisfies each particular axiom in the the meta-theory. But this is exactly parallel to the situation with sets and classes in ordinary ZFC, where we cannot generally even state that $L$ satisfies all of ZFC, only that it satisfies each particular axiom of ZFC expressible in the meta-theory.

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Also this and to a lesser extent, that. –  Asaf Karagila Apr 26 at 21:11
    
Yes, I see that the same issue has come up now numerous times! But it seems that there is variation in the precise question or motivation, and so the questions cannot really be said to be duplicates, even though the answers definitely overlap. –  Joel David Hamkins Apr 26 at 21:18
    
I agree, and I didn't suggest those as duplicates. Just as further entries to the list of related discussions. :-) –  Asaf Karagila Apr 26 at 21:31
    
Oh, I wasn't suggesting that you were! Rather, I meant only to express my own embarrassment at having evidently written about $V_\kappa\prec V$ so many times. –  Joel David Hamkins Apr 26 at 21:59
    
Thanks, nice answer. –  goblin Apr 26 at 22:23

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