@Fil Albuquerque,
There is a difference between not being able to classify a set class it might be a problem of language going on holidays and right out stating that the Set of all sets does not exist.
For starters, the Set of all Sets does not have to include a set that is undecidable
Clearly, if there are Sets, then:
Either all sets are the same size. Say same class of infinite. For those who believe in infinity.
Or some sets are bigger than others.
Stating as I did that there are the biggest Set possible boils down to:
Starting from an Axiom against actual infinities.
Starting from an Axiom that Sets have information size.