In ZF classes are used informally to resolve Russells Paradox, that is the collection of all sets that do not contain themselves does not form a set but a proper class. But doesn't the same paradox manifest itself when discussing the class of all classes that do not contain themselves?
The class of all classes not containing themselves
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0and the term 'heavy lifting'! – 2016-07-04
2 Answers
Classes in ZF are merely collections defined by a formula, that is $A=\{x\mid \varphi(x)\}$ for some formula $\varphi$.
It is obvious from this that every set is a class. However proper classes are not sets (as that would induce paradoxes). This means, in turn, that classes are not elements of other classes.
Thus discussion on "the classes of all classes that do not contain themselves" is essentially talking about sets again, which we already resolved.
Of course if you allow classes, and allow classes of classes (also known as hyper-classes or 2-classes) then the same logic applies you have have another level of a collection which you can define but is not an object of your universe.
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0Asaf, I have a couple of questions, I hope you don't mind. Firstly, would it be fair to say that the notion of a class make sense for any first-order structure, not just a model of set theory? For example, does it make sense to speak of the classes of $(\mathbb{N},0,S,+,\times)$ ? I'm thinking yes. Secondly, would it be fair to say that, since MK allows quantification over proper classes, thus some of its so called "classes" aren't really classes at all? – 2013-09-19
Von Neumann–Bernays–Gödel set theory is consistent and it's a theorem of Von Neumann–Bernays–Gödel set theory that there is no class of all classes that don't contain themselves. For any predicate describable in that theory, you can prove that there is a class of all sets satisfying that predicate but not necessarily prove that there is a class of all classes satisfying that predicate. You can describe the statement that no class contains itself and prove it but that's doesn't let you assert the existence of the class of all classes that don't contain themselves. In fact, in Von Neumann–Bernays–Gödel set theory, no class contains any proper class.
Source: https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory