A class can be about all things not in the class {Russell paradox} {Russell's paradox}, such as set of all sets that are not members of themselves.
sets
The set of all infinite sets is an infinite set and is a member of itself. The set of all sets is a member of itself. The set of all ideas can be an idea.
However, the set of all men is not a man. Therefore, sets with elements that are not classes cannot be members of themselves.
proof
Assume set of all sets that do not belong to themselves exists and is not a member of itself. Then it must belong to itself by its set definition, causing contradiction. If this set really does belong to itself, then it must not belong to itself by its set definition, causing contradiction. Therefore, set either does or does not belong to itself.
universal set
These two set types are mutually exclusive. The set of all sets that do not belong to themselves cannot be in either of these two set types. Therefore, no universal set exists.
class
Classes {class, set} have sets as members. Classes cannot be class members.
resolution
To resolve the paradox, replace the word "class" with the word "function" in paradox and proof.
Mathematical Sciences>Logic>Paradox
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Date Modified: 2022.0224