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Originally, I had written a whole spiel about the Von Neumann universe and Cantor's theorem in relation to the collection of objects that can be described in the language of set theory; but then I realised that the entire issue reduces to picking "some extension of ZFC with the words 'class' and 'size'" and a handful of logical arguments. So rather than try to address every detail, I'm simply going to list the arguments.

Premise 1-1

Every set is a class.

Premise 1-2

Every subset of a class is a set.

Premise 1-3

The powerset $\mathcal{P}(C)$ of a class $C$ contains all and only subsets of $C$.

Premise 1-4

$V$ is the class containing the empty set and all subsets of itself.

Conclusion 1

$\mathcal{P}(V)=V$

Premise 2-1

The size of the powerset $\mathcal{P}(C)$ of a class $C$ is $2^C$.

Premise 2-2

If $A=B$, then the size of $A$ is the same as that of $B$ ($A\cong B$)

Conclusion 2

$2^V\cong V$

Question

  1. What is the smallest class $E$ such that $V<E^V$.

  2. What is the smallest class $E$ such that $\kappa<E\implies \kappa^E\cong E$

Comment: The above is consistent with ZFC provided that the size of every cardinal is less than that of $V$. If every class which is contained in another class is a set, then the above is consistent with NBG.

Comment: The original question touched on $V=L$. Should there be a tag for that, or is it to specific?

R. Burton
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1 Answers1

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There are two notions of "power/function collection" at play here; your post focuses on one, but your particular question implicitly assumes something only true of the other. Untangling these we get two versions of your OP with correspondingly different answers.

Below I'll work in some theory $\mathsf{HCT}$ capable of handling sets, classes, and hyperclasses which is a conservative extension of $\mathsf{NBG}$; whipping up an appropriate theory is easy so I'll ignore the details. Analgously to $\mathsf{NBG}$, all objects in this theory will be hyperclasses, classes will be hyperclasses contained in some hyperclass, and sets will be classes contained in some class.

Let's look at the notion of "powercollection" first for simplicity. There are two ways we could try to define $\mathcal{P}(E)$ for $E$ a class - namely, as counting subsets or subclasses (and it's with respect to the latter that we need to bring hyperclass language into the picture):

For $E$ a class, let $\mathcal{P}_{small}(E)$ be the class of all sets which are subclasses of the original class, and let $\mathcal{P}_{large}(E)$ be the hyperclass of all classes which are subclasses of the original class.

When we look at function sets and think about the identification between $2^?$ and $\mathcal{P}(?)$, we run into an analogous "small vs. large" distinction. For a class $E$, the class maps $E\rightarrow 2$ corresponding to elements of $\mathcal{P}_{small}(E)$ are exactly those which are $0$ except on set-many elements. This suggests the following pair of definitions:

For $X,Y$ classes we let $X^{Y,small}$ (fixing some $x_0\in X$) be the hyperclass of class maps $f: Y\rightarrow X$ such that the class of $x\in X$ such that $f(x)\not=x_0$ is a set. We let $X^{Y,large}$ be the hyperclass of all class maps from $Y$ to $X$.

We then have:

  • Per Cantor, there is no surjection from $V$ to $\mathcal{P}_{large}(V)$, and there is a bijection between $2^{V,large}$ and $\mathcal{P}_{large}(V)$ so there is no surjection from $V$ to $2^{V,large}$ either.

  • On the other hand, we have $\mathcal{P}_{small}(V)=V$. Indeed, it's easy to build injections $V\rightarrow V^{V,small}$ and $V^{V,small}\rightarrow V$, so by Schroeder-Bernstein for classes we get a bijection between the two.

So under the "large" interpretation of your question your Conclusion $1$ is wrong and the answer to Question $1$ is $E=2$, and under the "small" interpretation of your question your Conclusion $1$ is true and the answer to Question $1$ is that there is no such $E$. Similarly, the answer to Question $2$ (restricting to proper class $E$s) is either "No such $E$ exists" or "Every $E$ satisfies the condition," respectively.

(Note that this exactly parallels the "set-level" situation for strong limit cardinals: if $\kappa$ is a strong limit cardinal, then there is a bijection between $V_\kappa$ and the set of subsets of $V_\kappa$ of size $<\kappa$, but $\mathcal{P}(V_\kappa)$ is strictly larger than $V_\kappa$. On that note, keep in mind that strong limit cardinals do provably exist in $\mathsf{ZFC}$.)

Noah Schweber
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  • The wording was quite deliberate - implicitly I was considering only the "small" case. I'm surprised to hear that $V^{V(small)}\cong V$, though; I would have expected $V$ to be the smallest $E$ such that $E^V\ncong V$. – R. Burton Aug 25 '20 at 23:35
  • As a side note, I'm not entirely sure what the benefit of hyperclasses would be. For one thing, there are trivial bijections between finite sets and finite hyperclasses - for example, $3\cong {V,\textbf{grp},\textbf{Top}}$. – R. Burton Aug 26 '20 at 00:09
  • More significantly, though if we allow collections of classes, why not collections of collections of classes and so forth? Following this line of thinking, we arrive at a hierarchy of "$n$-classes" that looks indistinguishable from a hierarchy of sets. I think back to this question - it seems that a theory of hyperclasses can prove all and only those things which a theory of sets can. – R. Burton Aug 26 '20 at 00:11
  • @R.Burton The main benefit of hyperclass theories is just the expanded language - they let you directly talk about e.g. the collection of all subclasses of a given class in a rigorous way. Note that this is also the primary value of class theories. There's a well-understood way to get a hierarchy of $n$-class theories which are conservative extensions of each other, so this is something we can do pretty freely. Why not allow hyperhyperclasses here? Well, in this particular case they're not relevant (yet). In general we work with what we obviously need. – Noah Schweber Aug 26 '20 at 00:14
  • @R.Burton As to "it seems that a theory of hyperclasses can prove all and only those things which a theory of sets can," sure, that's trivially true: given a hyperclass theory $T$, just consider the subtheory $T'$ consisting of all $T$-theorems about sets. Of course care has to be taken in going the other way: both $\mathsf{NBG}$ and $\mathsf{MK}$ are natural class theories, but only the former is conservative over $\mathsf{ZFC}$. (Meanwhile I'm not sure what the linked question has to do with any of this.) – Noah Schweber Aug 26 '20 at 00:16
  • And the fact that some hyperclasses are in bijection with sets is a total red herring: at no point did we assume that we cared about hyperclasses only because they're all "large." – Noah Schweber Aug 26 '20 at 00:18
  • @R.Burton Are there any further questions you have about this answer? – Noah Schweber Sep 16 '20 at 01:39
  • This works, but I still don't like it. It really seems to defeat the purpose of having classes to begin with. – R. Burton Sep 26 '20 at 13:45
  • @R.Burton What don't you like exactly? Keep in mind that classes aren't something we choose to have, they're something that is forced upon us: per Russell's paradox, any "collection theory" (set theory, class theory, hyperclass theory, ...) whatsoever will have formulas not corresponding to any object, and consequently properties of objects which naively should make sense for these "virtual objects" aren't even expressible in the object language. The conservative extension machinery, however, shows that this isn't a serious issue: we can always expressively extend our framework for free. – Noah Schweber Sep 26 '20 at 13:55
  • I'm reminded of a certain man being made to push a boulder up a hill. – R. Burton Sep 26 '20 at 13:59
  • @R.Burton OK, can you clarify why you're reminded of that? What's the concrete task you don't see being accomplished? – Noah Schweber Sep 26 '20 at 14:00
  • An upper bound on the size of models for "set-like" theories. Ideally, there ought to be some point at which further objects are no longer expressible in the language $\langle\in\rangle$ without introducing a contradiction (a "$0=1$" cardinal, if you will). I had thought that this would naturally be the class of sets. – R. Burton Sep 26 '20 at 14:05
  • @R.Burton First, we do have such upper bounds for any fixed set theory $T$: we need to enlarge our language to talk about "$T$-hyperclasses" directly. It's not even an issue of contradictions, the language simply doesn't support it. What we can't do is whip up an "ultimate" formalization of the naive idea of a collection. However, I tend to think of this as a positive phenomenon: "collection" is a non-formalizable idea which is mathematically useful! I find this wonderful. "Definition" is similar, with the Berry paradox and its relatives playing the role of Russell's paradox here. – Noah Schweber Sep 26 '20 at 14:08
  • (On the other hand, "algorithm" provides a surprising non-example per the Churc-Turing thesis.) Getting back to collection theories, we can clarify the "boundary" a bit. Any collection theory $T$ can quantify over its objects, and can talk about its "proper class analogues" individually, but can't even refer to individual "hyperclass analogues." So there is a concrete limit here: we lose quantification one level up and we lose everything two levels up. However, it may also help to mention the "setification" phenomenon: sometimes we can rephrase higher principles in object terms. – Noah Schweber Sep 26 '20 at 14:12
  • Consider e.g. the following definition of a measurable cardinal: "$\kappa$ is measurable iff there is a nontrivial elementary embedding of $V$ into some inner model with critical point $\kappa$." As written this isn't even expressible in $\mathsf{ZFC}$, we need to talk about classes. However, it is "setifiable:" any reasonable class theory like $\mathsf{NBG}$ can prove that $\kappa$ satisfies this definition iff there is a set which is a $\kappa$-complete ultrafilter on $\kappa$. Besides simplifying language, this process can lead to new ideas. Consider the Kunen inconsistency. – Noah Schweber Sep 26 '20 at 14:15
  • The Kunen inconsistency, a prior an $\mathsf{NBG}$ theorem, says that there is no nontrivial elementary embedding of $V$ into itself (that is, there is no Reinhardt cardinal). This can be "setified:" $\mathsf{ZFC}$ proves "There is no $\kappa$ such that there is a nontrivial elementary embedding of $V_{\kappa+2}$ into $V_{\kappa+2}$," and $\mathsf{NBG}$ proves that nontrivial e.e.s of $V$ into itself yield nontrivial e.e.s of $V_{\kappa+2}$ into itself for some $\kappa$. But the set version of the Kunen inconsistency suggests looking at "weak Reinhardt axioms:" – Noah Schweber Sep 26 '20 at 14:17
  • e.g. "There is some $\kappa$ such that there is a nontrivial e.e. of $V_{\kappa+1}$ into itself," or "... of $V_\kappa$ into itself," or more technical variants. These "rank-into-rank" axioms are really inspired by the set version of the Kunen inconsistency as opposed to the Kunen inconsistency itself, and are currently not known to be inconsistent with $\mathsf{ZFC}$. So setification is a nontrivial task (of course it isn't always possible) which in turn can have nontrivial impact. – Noah Schweber Sep 26 '20 at 14:20
  • Basically, via Russell's paradox a naive hope ("figure out how to talk about collections once and for all") is killed - but that paradox itself gives rise to new interesting phenomena. This isn't the only time this happens. For example, the Liar paradox imposes a limit on our ability to talk about truth, but the idea behind it leads to the incompleteness and undefinability theorems which are actually part of mathematics proper. As with many results in foundations of mathematics, whether this is bad or good depends on one's outlook. My stance is that impossibility results are starting points. – Noah Schweber Sep 26 '20 at 15:56