What kinds of operations are well-defined when working with sets, classes, conglomerates, and yet higher order collections?

Question

There are many foundations to set theory, ZFC, NBG, SEAR, to name a few, and while they differ in how sets, classes, and higher-order collections are represented as mathematical objects, they all attempt to provide useful tools to mathematicians without being glaringly inconsistent.

Note that many set theories stop at the set, or only weakly partially define classes. But for those that go further, I have the following question:

What kinds of operations are well-defined when working with sets, classes, conglomerates, and yet higher order collections? Or, what kinds of operations are commonly assumed to be well-defined in fields such as category theory, etc.?

Here are some operations which I am curious about.

1) Applying the axiom of choice to the class of all sets, or to the conglomerate of all classes, and so on.

2) Given two classes $A$ and $B$, forming the set of classes $\{A,B\}$. Similarly for sets of conglomerates, etc. Of course, the set of all classes should be disallowed by Russell's paradox.

3) Given two classes $A$ and $B$ and a set $C$, make a "set-class function" mapping $C$ into $A$, a "class-class function" mapping $A$ into $B$, and a "class-set function" mapping $A$ into $C$. Similarly for any pair of higher-order collections of any type.

4) Regarding the cardinality of a set, class, conglomerate (etc) as a way to make equivalence classes of sets, classes, conglomerates (etc), respectively.

5) Regarding the cardinality of sets, classes, conglomerates, and higher-order collections as a way to define an equivalence class among all mathematical objects.

6) Forming the set of all "class-class functions" (or is it a class?)

I'm the least confident that #'s 5-6 will be acceptable, but the rest seem reasonable to my untrained eyes.

Answer 1

This question is considered in detail in Mike Shulman's Set Theory for Category Theory.

The "big picture" way to think about it is to think that each time you do a power-set-type operation, you are constructing an object of a "higher order". You just then need to postulate high enough orders to do whatever it is you want. $U_0$ is the class of all sets, $U_1$ is the class of all classes of sets, etc.

Once you are forming classes of classes, etc., you are strictly outside what is provably consistent with ZFC, but set theorists routinely consider much, much stronger theories that do not seem to harbor contradiction. So it's probably okay.

Answer 2

Maybe too late, but only today I have seen this question posed 10 years ago.

Indexed families of (proper) classes can be legally defined in NBG: an indexed family of classes $(X_\alpha)_{\alpha\in A}$ is a subclass $X$ of the class $A\times \mathbf V$, where $\mathbf V$ is the class of all sets. More precisely, any subclass $X\subseteq A\times \mathbf V$ can be thought as an indexed family $(X_\alpha)_{\alpha\in A}$ of the classes $X_\alpha:=\{x\in \mathbf V:\langle \alpha,x\rangle\in X\}$.

So, a pair $(X,Y)$ of (proper) classes can be defined as the class $(\{0\}\times X)\cup(\{1\}\times Y)$, and such a pair has the property required for ordered pairs: two pairs of classes $(X,Y)$ and $(A,B)$ are equal if and only if $X=A$ and $Y=B$.

However, an argument similar to Russell's paradox shows that for any indexed family of classes $(X_\alpha)_{\alpha\in A}$ there exists a class $Y$ such that $Y\ne X_\alpha$ for all $\alpha\in A$. So, we cannot list all classes by an indexed family of classes. Yet the indexed family $(x)_{x\in\mathbf V}$ of all sets does exist in NBG.

Concerning the ``cardinality'' of an indexed family of classes $(X_\alpha)_{\alpha\in A}$, we can define it as the cardinal of the quotient class $A/R$ of $A$ by the equivalence relation $R:=\{\langle \alpha,\beta\rangle\in A\times A:X_\alpha=X_\beta\}$. The cardinal of a set $X$ is the smallest cardinal $\kappa$ for which there exists a bijective map $f:\kappa\to X$. The cardinal of a proper class is defined as the class $\mathbf{On}$ of all ordinals. The Axiom of Global Choice implies that for every proper class $X$ there exists a bijective function $F:\mathbf{On}\to X$.

Answer 3

There are many different ways of 'working with sets, classes, conglomerates, and yet higher order collections' -- you ask how each theory deals with these things, so I'll review some standard choices.

Grothendieck Universes

Probably the most prevalent is to use Grothendieck universes, named after Alexander Grothendieck but apparently first defined by Ernst Zermelo in 1930. For however many 'levels of collection' we want, subtract one and you've got the number of Grothendieck universes your theory needs -- if you only want one level (i.e. just sets) you can just work in ZFC or some other pure set theory, if you want sets and classes work in any of these theories augmented with an axiom asserting the existence of one Grothendieck universe, if you want conglomerates use an axiom asserting the existence of two -- as pointed out by Joel David Hamkins in an answer I can't find at the moment, there is a catch. We not only want one universe to contain the other, we want the smaller to be an elementary substructure of the larger, meaning it respects the smaller universes notion of truth regarding the elements of the smaller universe (so truths about sets don't change when we view them as classes).

Working in these theories, all higher order collections admit all operations you mention and output higher order collections of the same order as their largest input.

Inaccessible Cardinals

After this, probably the second most predominant option is working in whatever base pure set theory you like plus as many inaccessible cardinals as you would otherwise need Grothendieck universes; we then treat the cumulative hierarchy $V_\kappa$ up to each inaccessible $\kappa$ as the regimes of sets, classes, etc. These theories are equiconsistent as mentioned by Joel in this excellent answer (along with some lower consistency strength options that still serve many of the same purposes, mentioned at the end).

Accordingly, the same comment as above applies here.

Worldly Cardinals

For those concerned about consistency strength, a natural choice is a hierarchy of worldly cardinals in place of inaccessible cardinals -- these are cardinals $\kappa$ such that $V_\kappa\models ZFC$, something implied by inaccessibility but not necessarily implying it. Here, we use the cumulative hierarchy as above to form each regime of collection.

Since each regime here still models all of $ZFC$, once again all operations you ask about are permissible at each level without issue.

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Here I believe we begin to trail off from the 'standard options'; for most people who need to consider multiple levels of collection, one of the above options (or the lower consistency strength ones mentioned in Joel's excellent answer linked above) serves the purpose well enough. For people who prefer a theory that deals with multiple levels of collection directly at the primitive level, I uploaded a short note to the arxiv last year laying out one such theory, and there are many others I should link here (and will on a night when I'm more well rested).

Answer 4

Regarding ZFC, I don't see any way to make sense of your questions; the language of ZFC doesn't allow you to mention anything called a "class", a "conglomerate" or a "higher order collection".