Full Idea
By the mid 1890s Cantor was aware that there could be no set of all sets, as its cardinal number would have to be the largest cardinal number, while his own theorem shows that for any cardinal there is a larger.
Gist of Idea
For any cardinal there is always a larger one (so there is no set of all sets)
Source
Penelope Maddy (Naturalism in Mathematics [1997], I.1)
Book Reference
Maddy,Penelope: 'Naturalism in Mathematics' [OUP 2000], p.17
A Reaction
There is always a larger cardinal because of the power set axiom. Some people regard that with suspicion.