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Why are there no big cardinals beyond the universal set?

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though unpopular, “minimal” explanation for why there is no set of all sets is that the supposition that there is .... though we can derive that the universal set doesn't exist from Foundation, we have to provide that “deeper” fact in ...... also Linnebo (2013, 207f.). 34 See Koellner (2011) for more on the Large Cardinal Hierarchy. read more

Yes, you can extend the notion of cardinality to classes, and it is consistent that there is only one cardinal for classes (e.g. $V=L$ implies that all classes have the same size), or it is consistent that there are classes which are incomparable (e.g. if you add two Cohen subsets to a proper class of cardinals, the class of the pairs cannot be definably well-ordered and it is incomparable in size with the class of ordinals). read more

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Why is the universe of sets not a set?
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