Because S is a set (and not a proper class like the class of ordinal numbers), this construction ultimately must stop with a maximal member of S.
I saw how my parents made sure that my younger brother was placed in the proper class; it was common knowledge that some schools were quick to place your black child — particularly your black male child — in special-education classes.
There were strong anarchist streaks in the New Left of the 1960s — stronger than the socialist streak, in fact, despite all the work Marxists did to define proper class categories for the student movement.
But enough stars turned up to ensure a reasonable quotient of glitz – among them Johnny Depp, Kristen Stewart, Tilda Swinton, Ralph Fiennes, Juliette Binoche, Mark Ruffalo, the inevitable Shia LaBeouf, and adding some proper class, the Nobel prizewinning novelist Orhan Pamuk.
Some logicians use the terms "class" and "set" interchangeably; others distinguish between them, defining a set (for example) as a class that is itself a member of some class and defining a proper class as one that is not a member of any class.
So Ω + 1 must exist, and is a proper class with the proper class Ω as an element.
This process must eventually terminate, since otherwise the union of the chains so generated would constitute a proper class, making \(P\) itself a proper class contrary to assumption.
A proper class is defined as a class which is not a set.
Any mathematical object whatsoever can always be viewed as a set, or a proper class.
Thus, the class of all cardinals is not a set, but a proper class.
(It's certainly not a formula false in a proper class structure).
