User blog comment:LittlePeng9/Higher order set theory/@comment-2033667-20141010133022

okay we need to name these things. the general term for a collection appearing anywhere in this hierarchy of sets and classes and stuff is an "oodle." like sets and classes, oodles can have members which are other oodles, or possibly ur-elements depending on the model we want to use (but in this context they'll just contain other oodles). given two oodles A and B, if every element of A is an element of B, A is a suboodle of B and B is a superoodle of A. the poweroodle of an oodle A is the oodle containing every suboodle of A.

we also define some names:


 * sets are 0-classes
 * a class (or 1-class) is an oodle of sets
 * a superclass (or 2-class) is an oodle of classes
 * a supersuperclass (or 3-class) is an oodle of superclasses, etc.
 * a megaclass (or w-class) is an oodle containing sets, classes, superclasses, supersuperclasses, etc.

oodle is meant to describe EVERY kind of collection that could possibly appear in the above blog post, so we never run out of this terminology. as such please resist the temptation to define things such as "1-oodle," "2-oodle," "superoodle," etc. oodle is meant to be inexhaustible.