Meaning of "There exists a proper class of..."

Question

How a statement of the form "There exists a proper class of..." can be formalized in $\sf ZFC$? It sounds a bit like an oxymoron to me, because it's the very essense of a proper class that it does not exist, only its elements (as identified by some predicate) exist.

I've never seen a definition that would assign a formal meaning to statements of this form, but my understanding that I inferred from context is as follows:

Let $P(x)$ be a predicate with a free variable $x$. The statement "There exists a proper class of objects satisfying $P$" means $\nexists y\,\forall x((x\in y)\iff P(x))$ (i.e. "There is no set containing exactly those objects satisfying $P$").

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Does this understanding match a commonly accepted one?

Answer 1

Mathematics is communicated in natural language, ultimately.

This means that we write phrases which might not be formally correct, but are well understood by everyone who is familiar with the context.

"There exists a proper class of $\varphi$" means that the collection $\{x\mid \varphi(x)\}$ is a proper class. Certainly this collection is definable, and when we say "there exists a proper class" we mean to say that there does not exists $A$ such that the collection defined by $\varphi$ is a subset of $A$.

So when we say that there exists a proper class of singletons we mean that there is no set of all singletons; and there exists a proper class of vector spaces over $\Bbb R$ we mean there is no set of all vector spaces over $\Bbb R$.

Answer 2

Usually we talk about the proper class of something well ordered such as cardinals. In which case you can substitute unbounded, $$\forall \kappa \exists \lambda > \kappa P(\lambda)$$ means that there is a proper class of cardinals satisfying $P$. In general you can use the cumulative hierarchy, $$\forall \kappa \exists x( \text{Rank}(x) > \kappa \wedge P(x))$$ says that there is a proper class of sets $x$ satisfying $P$.