I'm currently working on some universal algebra using proper classes (in MK class theory), and I repeatedly run into situations where I want to collect together some proper classes as the members of a new algebraic structure.

For example, the construction given here yields a bunch of equivalence classes $\equiv/(m_0,m_1)$ for $(m_0,m_1)\in\mathbb{M}^2$, and if $\mathbb{M}$ is a proper class then each of these equivalence classes is a proper class. Despite this we would like to treat $\mathbb{M}^2/\equiv$ as a group, so to get around the problem we can technically collect together one representative from each equivalence class and be good to go with the class of representatives serving as the new group.

This feels clunky to me from a logical standpoint every time I have to do it, and it makes me wonder if there may be situations where we want to collect up some proper classes which do not admit representatives so simply. Consequently, I would like to add an axiom to MK which allows me to dictate one of two things:

  1. I can collect together proper classes into 'hyper classes' under certain circumstances. This seems like the direct and obvious route at first; it is suggested by Andreas Blass under similar circumstances, however this would require an extension of the formal language of MK to include a relation symbol between proper classes and hyper classes that is not $\in$, and this bothers me. (this may bother me less as I learn more logic/model theory)

  2. Classes which are definable as equivalence classes using the first order language of sets and a relation $\mathcal{R}$ on a class $\mathbb{B}$ (possibly a proper class) are sets.

The second one is more along the lines of what I want, but this form of it is inconsistent. For example, if we let $\mathfrak{G}(O_n)$ be the Grothendieck ring of the ordinals as defined here for $\omega_1$ (just replace $\omega_1$ with $O_n$ in the construction), we technically need to take a quotient of $\mathfrak{G}(O_n)$ by the equivalence relation $$\equiv=\{(\alpha-\beta,\gamma-\zeta):\alpha+\zeta=\gamma+\beta\}$$ to remove extra 'difference representations' of the same element in $\mathfrak{G}(O_n)$. We now have that $\equiv/(\alpha-\beta)$ is a set for any choice of $\alpha,\beta$ under this new axiom, but $$\equiv/(0-0)=\{\alpha-\alpha:\alpha\in O_n\}$$ is trivially in bijection with $O_n$, a proper class.

In summary, my question is:

What is a concise axiom that (when added to MK class theory) would allow us to collect together proper classes satisfying certain algebraic formulae as though they were sets?

If an extension of the language of sets with an additional relation for hyper classes seems the most concise option, I would be open to an argument in favor of that as well.

It also seems from the discussion here that I may want to simply consider collecting these objects together in the meta-theory, but I do not know what pitfalls await me if I try this apparently bold route.

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    $\begingroup$ The term is either hyperclass, 2-class, or conglomerate. Superclass is just a larger class ($V$ is a superclass of $\rm Ord$). $\endgroup$ – Asaf Karagila Feb 7 '18 at 8:42
  • $\begingroup$ @AsafKaragila Thank you, I'll edit. $\endgroup$ – Alec Rhea Feb 7 '18 at 18:53

Really, what you want to have is a set-theoretic structure on top of the universe, with classes and meta-classes and hyper-classes and so on, in a set-theoretic realm continuing to build into ranks above the ordinals.

One can make this precise by talking about classes coding these meta-class structures. The process is referred to as unrolling, and goes back to Marek and Mostowski.

The basic idea is to code these higher-order classes with well-founded class relations on the ordinals, in much the same way that one can code hereditarily countable objects by means of a well-founded relation on the natural numbers. One considers the family of such codes and then defines the corresponding natural element-of relation on the codes and the natural equivalence relation, which in the right theory is a congruence with respect to that element-of relation.

If you start with a model of KMCC, which is Kelley-Morse set theory with the class-collection principle (this is strictly stronger than KM by a result of mine and Victoria Gitman's), then you can build a model of $\text{ZFC}^-_I$, which is ZFC without the power set axiom, but with a largest cardinal, which is inaccessible. Basically, Ord itself turns into the largest cardinal of the higher realm, and you get $\text{ZFC}^-$ for the rest of the higher-order structure.

Indeed, those two theories are bi-interpretable in a precise sense, and this is the answer to your question: if you have KMCC, then you can just jump into the unrolled universe, where the higher-order classes exist in a $\text{ZFC}^-$ context and the old Ord becomes an inaccessible cardinal and the largest cardinal.

My student Kameryn Williams is writing a chapter in his dissertation about figuring out exactly what set-theoretic strength you need in order to have exactly which theories in the unrolled structure. For example, you basically need the principle of elementary transfinite recursion (ETR) in order for basic facts about the unrolling process to work out, and he has some level-by-level results about how the strength of second-order set theory in the base model is revealed by increased strength of set theory in the unrolled structure. I'll encourage him to post further information.

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    $\begingroup$ This seems like the best solution if the desired hyperclasses are only "class-sized" (analogously to the "hereditarily countable" requirement for unrolling as reals). So it should solve the OP's difficulties. If one wants "full-sized" hyperclasses, then MK won't suffice. $\endgroup$ – Andreas Blass Feb 7 '18 at 19:48
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    $\begingroup$ Yes, I agree completely. In that case, one wants to be working in a ZFC set theory with universes of some kind, such as a proper class of inaccessible cardinals, or possibly Ackermann set theory. $\endgroup$ – Joel David Hamkins Feb 7 '18 at 19:57
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    $\begingroup$ You can find a little about ETR in my paper on class games with Victoria Gitman, and also in my work with several co-authors on the class forcing theorem. The work on KMCC is not yet available. Kameryn's dissertation results will be available in a few months if not sooner. $\endgroup$ – Joel David Hamkins Feb 7 '18 at 20:51
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    $\begingroup$ You can pick representatives from 'full-sized' hyperclasses under some extra assumptions. If your unrolled structure is of the form $L[A]$—which, by work of Antos and Friedman (arxiv.org/abs/1510.04082), can be forced over MK + Class Collection + the Ord-Dependent Choice schema—then you have a definable global well-order of the unrolled structure, which you can use to make the choices. And the MK + ... theory is equiconsistent with MK, so these extra assumptions aren't costly. $\endgroup$ – Kameryn Williams Feb 7 '18 at 20:54
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    $\begingroup$ The chapter Joel mentioned isn't quite to the point of being publicly shared around, but I can post a link to it within a few months. Alternatively, feel free to contact me privately. In brief, while ETR is enough to get that the unrolling satisfies the basic axioms of set theory, the resulting structure has a fairly weak theory. Most likely, you just want to work with KMCC, and not sweat the small difference in consistency strength. $\endgroup$ – Kameryn Williams Feb 7 '18 at 21:28

Unless you are committed to beginning with MK class theory, which is not conservative over ZFC, I suspect you can get all you want working in Ackermann's set theory as developed by W. Reinhardt in Ackermann's set theory equals ZF, Ann of Math Log 2, pp. 189-249. There is a nice overview of the theory by Azriel Levy in The Role of Classes in Set Theory, which appears both as chapter of Foundations of Set Theory (Second Revised Edition), A. Fraenkel, Y. Bar-Hillel and A. Levy, North-Holland Publishing Co. (1973) and as a chapter of Sets and Classes (G.H. Müller ed), North-Holland Publishing Co. (1976).

In Reinhardt's version of Ackermann's theory, which is conservative over ZFC (as well as over NBG with Global Choice), given a class $A$ having the power of $On$ one can form $P(A), PP(A), PPP(A), ...$, where $P(A)$ is the power class of $A$.

For some reason Ackermann's theory has not received much attention. Perhaps Joel or someone else knowledgable about such matters can explain why this has been the case.

Edit. I wrote this before I saw Joel's comment.

  • $\begingroup$ I was just about to ask for a reference on Ackermann's set theory, much appreciated! As for the coincidence, there is a saying about great minds for precisely this situation :). $\endgroup$ – Alec Rhea Feb 7 '18 at 20:15
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    $\begingroup$ I think many set theorists find Ackermann's set theory confusing, in part because of how $V$ is not everything for Ackermann and the universe continues above $V$. So the problem with it is simply a terminological conflict with the established set-theoretic conventions. It seems to me that the theory ZFC plus $V_\kappa\prec V$, a fully correct cardinal, amounts to basically the same thing, where $V_\kappa$ plays the role of what Ackermann refers to as $V$. This fully correct cardinal way of understanding things is commonly used, and I view it as what Ackermann set theory has turned into. $\endgroup$ – Joel David Hamkins Feb 8 '18 at 14:15

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