Why do we need classes to list all cardinal numbers?

Question

In E. Kamke's Theory of sets he shows that for any set $S$ of cardinal numbers, there exists some cardinal number that is strictly greater than all cardinal numbers in $S$. Thus "the set of all cardinal numbers" doesn't make sense. This I understand. I did some more reading and found out that to list all the cardinal numbers you need a class.

So my questions are

(a) What is a class and how does it differ from a set?

(b) Why does it allow us to list all the cardinal numbers?

Thanks.

Answer 1

In mainstream set theory (that is, based on the ZFC axioms), a "class" means a group/collection/plurality/whatever of sets that we can identify by a property they all share, but which may not constitute the elements of a set.

Classes differ from sets by:

1. A class cannot be a member of a set (or of another class).
2. Classes do not actually exist in the formal development of (ZFC) set theory. Speaking of a class is just a vivid way of speaking of the property (say, as a logical formula) that defines membership in the class.

Thus we can speak of the class of all sets (where the property is just "is a set"), or of a class whose defining property is "is a cardinal number'.

Answer 2

Intuitively, given any formula $\varphi$ with one free variable, we might want to reason about the collection $\{ x \mid \varphi(x) \}$ of objects $x$ that satisfy $\varphi(x)$. We'd like this to be 'the set of all $x$ such that $\varphi(x)$', but unfortunately that leads to Russell's paradox. So, to be safe, we—informally—refer to $X = \{ x \mid \varphi(x) \}$ as a class. Formally the word 'class' is meaningless on its own and $X$ is not an object that we can reason about within the usual set theory; when we write '$x \in X$', what we really mean is '$\varphi(x)$'.

To say that a class $X = \{ x \mid \varphi(x) \}$ is a set is to say that there exists a set whose elements are precisely those objects $x$ such that $\varphi(x)$ is true. In this case, we can then treat $X$ as we treat any other set.

It's possible to define a formula in ZFC that says '$x$ is a cardinal number', and so there is a class $\mathsf{Card} = \{ x \mid x \text{ is a cardinal} \}$. However, this class is too big to be a set—this is proved by showing that, if it were a set, then it would have a cardinality (since all sets have cardinalities), and this cardinality would be greater than all cardinalities (which is a contradiction, since it would then be larger than itself).