# First-order set theory : What is the class of all sets in ZFC?

What do we mean when we let the universal set be the class of all sets? How do I intuitively think about this? Do I just think of it as a collection of all sets? Also, is the Axiom of Regularity a part of ZFC?

I have to prove a statement in the class of all sets, but I'm not sure what I'm supposed to assume since I've no idea what a class is (besides the definition 'a collection which is not a set because it causes paradoxes i.e Russel's paradox)

• A class is the "meaning" of a predicate (e.g. a formula with a free variable $\varphi(x)$). The "trick" of $\mathsf {ZF}$ (due to Russell's paradox) is that not every predicate define a set. Oct 3, 2017 at 6:11
• Thus, the "uiversal class" is the "meaning" (or extension) of the predicate $x=x$ (and we know it is not a set). Oct 3, 2017 at 6:12
• There are several threads dealing with universal sets, or the difference between sets and classes, or what are sets. It seems that you have something specific that you want to understand, but you're beating around the bush to "try and be more general", which turns out just "more confused". If you tell us exactly what's the issue, then it is easier to understand what exactly is unclear. Oct 3, 2017 at 7:57
• See also Yiannis Moschovakis, Notes on set theory , Springer, Ch.3.19 Classes. Oct 4, 2017 at 8:19

Intuitively, a class is a collection of sets. It may be a set, but some collections are not sets. The class of all sets and the class of all ordinals are examples of classes which are not sets and are called proper classes. Intuitively, all proper classes have the "same cardinality as the universe" because anything smaller can be bijected with an ordinal and shown to be a set using replacement. A common way to prove something about all sets is by transfinite induction on the rank of the set. You prove it is true for the empty set, then prove that if it is true for all sets of a given rank it is true for all sets of the next rank. For limit ranks you need to prove that if it is true for all lower ranks it is true for the limit.

The following is meant to be a clarification of Ross Millikan's post:

In most commonly used set theories, we define classes (sometimes known as virtual classes) as a collection of sets satisfying some property. More precisely: Let $\phi(x, y_1, \ldots, y_n)$ be a formula in the language of set theory and let $p_1, \ldots, p_n$ be sets (parameters). Then $$C = \{x \mid \phi(x, p_1, \ldots, p_n \}$$ is a (virtual) class and, at least in $\mathrm{ZFC}$, all classes are of this form - for varying $\phi$ and $p_1, \ldots, p_n$.

As an example consider $V$ - the class of all sets. $V$ is a class in the above sense since $$V = \{ x \mid x = x \}.$$ As another example consider $P$ - the class of all pairs. $P$ is a class since $$P = \{ x \mid \exists y \exists z \colon x = (y,z) \}.$$

We can use $P$ to form another class, the class $X$ of all pairs whose first coordinate is a natural number: \begin{align*} X & = \{ x \mid x = (y,z) \in P \wedge y \in \mathbb N \} \\ &= \{ x \mid \exists y \exists z \colon x = (y,z) \wedge y \in \mathbb N \}. \end{align*}

It's also useful to note that given two classes $A,B$ the intersection of those two - call it $A \cap B$ - is a class. Here is why:

Fix formulas $\phi, \chi$ and parameters $p_1, \ldots, p_m, q_1, \ldots, q_n$ such that $$A = \{x \mid \phi(x, p_1, \ldots, p_m) \},$$ $$B = \{x \mid \chi(x, q_1, \ldots, q_n) \}.$$ Then $$A \cap B = \{ x \mid \phi(x, p_1, \ldots, p_m) \wedge \chi(x, q_1, \ldots, q_n) \}.$$

And, as you might imagine, there are many more ways in which we can combine known classes in order to generate new ones.

• Thank you for the answer. The examples, in particular, were very helpful.
– John
Oct 3, 2017 at 18:43
• You're very welcome! Oct 3, 2017 at 18:51

In much of classical set theory the idea of a 'class' is not formally defined. Intuitively it is a collection of sets, and we can say that a class contains a particular set or that it does not contain it. However, we cannot manipulate classes the way that we manipulate sets. In set theory, for example, we have an axiom of specification that allows us to produce a new (sub)set from any given set by specifying elements satisfying certain properties. This is not the case for a class -- we do not allow the creation of new classes from old classes via such specification, as Russell's paradoxes suggest that such entities need not be logically consistent. So perhaps one might think of a class as a very limited set that you can't play around with.

• What does 'the class of all sets' specifically refer to?
– John
Oct 3, 2017 at 5:24
• I disagree with your answer in all points. Classes are usually defined in any treatment of axiomatic set theory, i.e. they are defined whenever a framework for sets themselves is defined. And in all those cases we do specifically allow the formation of new classes from old classes. Oct 3, 2017 at 7:39