For experts who work in ZFC, it is common knowledge that one cannot in general define a countable intersection/union of proper classes. However, in my work as a ring theorist I intersect infinite collections of proper classes all the time.

Here is a simple example. Let a ring $R$ be called *$n$-Dedekind-finite* if $ab=1\implies ba=1$ for $a,b\in \mathbb{M}_n(R)$ (the $n\times n$ matrix ring over $R$). A ring $R$ is said to be *stably finite* if it is $n$-Dedekind-finite for every integer $n\geq 1$. The class of stably finite rings is the intersection (over positive integers $n$) of the classes of $n$-Dedekind-finite rings.

So my question is a straightforward one. What principle makes it okay for me to intersect classes in this manner?

Phrased a little differently my question is the following. Suppose we have an $\mathbb{N}$-indexed collection of sentences $S_n$ in the first-order language of rings. Why is it valid (in my day-to-day work) to form the class of rings satisfying $\land_{n\in \mathbb{N}}S_n$, even though the corresponding class doesn't necessarily exist if each $S_n$ is a sentence in the language of ZFC?

[Part of my motivation for this question is the idea that much of "normal mathematics" can be done in ZFC. That's how I've viewed my own work. I'm happy to think of my rings living in $V$, subject to all the constraints of ZFC. (In some cases I might go further, by invoking the existence of universes or the continuum hypothesis, but that is not the norm.) But I'm unsure how to justify my use of infinite Boolean operations on proper classes. Especially since, in some cases, my conditions on the rings are conditions on sets, from the language of ZFC!]

**Another way to think about all of this:**

Looking at the (currently three) answers to this question, it appears that people took my question to be asking about definability rather than existence. Under that interpretation of my question the answers are spot on, and I appreciate what they are saying.

Moreover, those three answers helped me realize that my question was more along the lines of the MathOverflow question "The set of Godel numbers of true sentences" as interpreted by Andreas Blass, from a Platonistic viewpoint, about *existence* rather than *definability*. So, I suppose that the answer I was looking for was something like:

Pace, you were **wrong** to say that the intersection "doesn't necessarily exist". It exists as a subcollection $S$ of the Platonistic universe $V$. You just cannot prove (or even properly *state*) in ZFC the fact that the class $S$ equals the intersection you are looking at. The "principle that makes it okay for [you] to intersect classes in this manner" is your personal Platonistic view of the universe of true sets, not any sort of logical principle that can be stated in a first-order way.

Any additional thoughts people have on this are welcome.

internalto the universe of sets. Not external. $\endgroup$ – Asaf Karagila Jan 30 '19 at 21:10