Trying to understand the history behind Russell's Paradox

[GrandizerII]

I think I get it now. The one-to-one mapping is done randomly, not based on any specific set of guidelines. When an entity is mapped to a subclass, it either is mapped to a subclass that contains it or it's mapped to a subclass that does not contain it. Sometimes it's possible that all entities are each mapped to a subclass that contains it, in which case it will not be possible to show that the one-to-one mapping f is impossible. Other times, however, some of the entities will be mapped to subclasses that don't respectively contain them, in which case it is then possible to show how the mapping f is impossible (because in this case one of the subclasses could not logically map back on to any of the entities, which means the number of the subclasses is greater than the number of the corresponding initial entities).

Did I get this right?