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I noted that issues such as Russell's Paradox involving the set of all sets that don't contain themselves can be resolved by stating that the object that is all set that don't contain themselves is a proper class.

But then you can generate the same issue by talking about say the class of all classes that don't contain themselves. Etc...

From here I come to the conclusion that there is an infinite hierarchy of such set-like generalizes

$$ C_0, C_1 .... $$

Where $C_0$ denotes a set, $C_1$ denotes a class, etc...

My question is, has anyone found any interesting behavior with the upper level objects. Ie is there any interesting structure that is unique about higher order classes compared to lower level ones.

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    $\begingroup$ Words to live by: hyperclasses and 2-classes. $\endgroup$ – Asaf Karagila Jul 19 '15 at 3:55
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    $\begingroup$ Relevant. $\endgroup$ – Henning Makholm Jul 19 '15 at 4:03
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    $\begingroup$ Also, that's how you essentially end up with a rudimentary type theory. And possible, ants. $\endgroup$ – Asaf Karagila Jul 19 '15 at 4:07
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    $\begingroup$ @Henning: You had me with transfinite fruit hierarchy. :-) $\endgroup$ – Brian M. Scott Jul 19 '15 at 4:48

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