I am still struggling with the concept of classes in ZF set theory. From Jech, Set Theory, p.5:
For every formula $\varphi(x)$ there is a class (which is definable). There are countably many definable classes.
For every formula $\varphi(x,y_1,\dots,y_n)$ and every tuple of sets $p_1,\dots,p_n$ there is a class (which is definable, when the sets $p_1,\dots,p_n$ are definable). There are "countably many times class-many", i.e. class-many such classes.
My questions are:
Can the concept of "arbitrary collections" of sets be sensibly conceived that are not even classes, i.e. not given by a formula + parameters?
(Possible answer: No, because we cannot talk about them.)
Can the classes of NGB set theory be conceived as "arbitrary collections"?
If we consider sets (of ZF set theory) as the base type ("type 0"), there are three candidates for "type 1" as collections of sets:
- sets of sets (collections that are co-extensive with sets)
- classes of sets (collections that are given by a formula/property)
- arbitrary collections of sets
Have all of these "type theories" been worked out, or do some of them simply not work?