What should we call the 'sets' which don't exist under certain set theory axioms?

Question

For example we know that the set of all ordinals does not exist in ZFC, so what should we call it? Set? Collection?

Answer 1

The usual term is proper class.

$\color{red}{\textbf{Warning:}}$

Beware of thinking about proper classes as objects. No model of $\mathsf{ZF(C)}$ has a proper class as an object, and so any arguments which treat them as sets are informal.

If $C = \{ x : \phi(x) \}$ is a class, it's common to write $y \in C$ when what we really mean is $\phi(y)$. So when we write $\alpha \in \mathbf{On}$ (or similar) to denote the fact that $\alpha$ is an ordinal, what we mean is that $\alpha$ satisfies a particular formula that defines all ordinals (and only ordinals).

There are set theories that have proper classes as objects, e.g. $\mathsf{NBG}$, in which case you can treat them as objects which have members; but in $\mathsf{ZF(C)}$ you can't do this.

Note in particular that, in $\mathsf{ZF(C)}$ $$\{ x \, :\, x \not \in x \} = V$$ where $V$ is the 'universe', the proper class of all sets. But we don't get the contradiction of Russell's paradox: the question of $V \in V$ or $V \not \in V$ does not arise because $V$ is not an object in the model.

Answer 2

In ZF if $\varphi(x,p)$ is a formula, then $\{x\mid\varphi(x,p)\}$ is called a class, where $p$ is a fixed parameter (which can vary between the classes, of course).

The parameters can be sets from the universe, which imply that every set is a class simply by the formula $\varphi(x,p):= x\in p$, but there are classes which are not sets. These classes are called proper classes. The universe is a proper class, as is the class of singletons, or ordinals, and so on.

We can still address proper classes in proofs because we have a defining formula, so we can say "For every $x$, if $\varphi(x)$ holds then ..." which is to say that if $x$ in that and that class, then we infer some properties of $x$.

But it is important to understand that proper classes are merely syntactical constructs which allow us to say something on a definable collection of elements. If one wishes to have proper classes as elements of the universe then one can use two-sorted theories which extend ZF(C), NBG and MK are such theories. In these theories we have two types of objects, classes and sets, and this allows us to quantify over classes and have them as actual objects, but even then not all classes are sets.

---

To read more:

1. Does $A=\{a|\forall x\in \emptyset\ H(x,a) \}$ make sense?
2. How do we justify functions on the Ordinals
3. difference between class, set , family and collection
4. What can I do with proper classes?
5. Difference between a class and a set