Talk:Proper class

"because that would cause a paradox"

This is a description that has popped up on GS occasionally, but I think it may mislead a bit about "why" classes are proper. If you take the class of infinite cardinals at which GCH fails, by Easton's theorem it's consistent that this is the empty set, and it's also consistent that it can be a proper class. Since it's consistently a set, there must be no paradox/(contradiction with ZFC) that makes it a proper class, yet there are valid models in which ZFC holds and it is a proper class. (This example is given by Asaf Karagila here). Additionally, no proper class ever contains another proper class (since over ZFC, proper classes are wrappers for formulae), so the second situation cannot happen in order to show that a class is proper.

AFAIK proper classes are exactly the classes that contain elements of unbounded rank (i.e. a class \(C\) is proper iff for any ordinal \(\alpha\), there exists a \(b\in C\) such that \(\textrm{rank}(b)>\alpha\)), but I don't think this can be stated in ZFC. If this holds it may be seen as justification for the common phrase "too large to be a set". C7X (talk) 23:15, 31 August 2023 (UTC)