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There is a well-known hierarchy of infinite cardinalities for sets. I've heard it said that proper classes are from a certain point of view "too large" to be sets.

Are some proper classes larger than others, in some set-theoretical system or other? Or are they all the same size or all incomparable in systems extending ZFC to talk about classes?

I suppose the motivation is: Is it possible to think about larger and larger classes? And have a class of all classes "too large" to be a class, call it a "collection", etc?

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  • $\begingroup$ I was meaning to ask whether something similar can be done for proper classes. And whether the class of all "cardinalities" of classes is not a class? $\endgroup$ May 9, 2013 at 18:15
  • $\begingroup$ @Frank: Why $2^{\aleph_0}=\aleph_1$? Why the next set has size $\aleph_2$? Do you assume $\sf GCH$? $\endgroup$
    – Asaf Karagila
    May 9, 2013 at 18:15
  • $\begingroup$ You may want to take a look at mathoverflow.net/questions/44303/cardinality-of-classes where a similar question was asked. Also, Andreas Blass has some interesting comments you may find useful as well, see for example: mathoverflow.net/questions/115091/… $\endgroup$ May 9, 2013 at 18:20
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    $\begingroup$ @Frank: I find it of utter importance to remark when using things like $\sf GCH$. Otherwise you help perpetuate mistakes like "the cardinality of the continuum is $\aleph_1$ by definition". There is no wrong in saying something like "Assume $\sf GCH$ then ...". $\endgroup$
    – Asaf Karagila
    May 9, 2013 at 18:30
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    $\begingroup$ If we use surjections to compare sizes, one limitation on the kind of incomparability that you mention is that the class of ordinals is a surjective image of every proper class (consider the values of the rank function on the class.) $\endgroup$ May 9, 2013 at 22:12

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It is consistent with $\sf ZFC$ that every two proper classes have a bijection between them. That is to say, if $A$ and $B$ are two proper classes then there is a class $C$ which is a class of ordered pairs which is a bijection from $A$ onto $B$.

For example, if we assume that $V=L$, then this is true.

But it is also consistent that this is not the case, and for example there is no bijection between $\sf Ord$, the class of ordinals, and $V$, the class of all sets.

For your motivation, do note that in $\sf ZFC$, or even in class-able set theories like $\sf NBG$ or $\sf MK$ which extend $\sf ZFC$, one cannot talk about classes of proper classes.

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