Full Idea
The realist meets the Burali-Forti paradox by saying that all the ordinals are a 'class', not a set. A proper class is what we discuss when we say "all" the so-and-sos when they cannot be reached by normal set-construction. Grammar is their only limit.
Clarification
For the paradox, see Idea 8674
Gist of Idea
Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets'
Source
Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3)
Book Reference
Friend,Michèle: 'Introducing the Philosophy of Mathematics' [Acumen 2007], p.27
A Reaction
This strategy would be useful for Class Nominalism, which tries to define properties in terms of classes, but gets tangled in paradoxes. But why bother with strict sets if easy-going classes will do just as well? Descartes's Dream: everything is rational.