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.
Best Answer
You will have to come up with trickier examples.
There is a formula $\phi(R, n)$ of ZFC with parameters $R$ and $n$ whose meaning is "$R$ is an $n$-Dedekind finite ring". One might think that $n$ is a "meta-level" number, i.e., a number which is "outside" of set theory. That is not how ordinary mathematics works. We take set theory as the ambient universe in which everything happens. The natural way of reading the definition you gave is that it all happens inside set theory, and that $n$ denotes an element of a certain set $\mathbb{N}$. If one were to play tricks with logic, one could of course consider a situation in which $n$ is a meta-level number outside of set theory, but that is not how mathematics is done. Only logicians do such things.
The class of all stably finite rings is simply $\{R \mid \forall n \geq 1 \,.\, \phi(R, n)\}$. In other words, the imagined intersection of a countable collection of classes is not needed at all, because $$\bigcap_{n \geq 1} \{R \mid \phi(R, n)\} = \{R \mid \forall n \geq 1 \,.\, \phi(R, n)\}.$$ The point is that the property of being $n$-Dedekind finite is uniform in $n$ and so a single formula $\phi(n, R)$ works. The situation would be different if you proposed an infinite sequence of classes $$C_1, C_2, C_3, \ldots$$ indexed by meta-level natural numbers, where each $C_i$ is defined by some formula $\psi_i$, so that $C_i = \{x \mid \psi_i(x)\}$. Furthermore, it would have to be the case that the formulas $\psi_i$ are non-uniform, i.e., there is no single formula $\Psi$ such that $\psi_i(x) \Leftrightarrow \Psi(\overline{i},x)$. Here $i$ is a meta-level numbers and $\overline{i}$ is the element of the set $\mathbb{N}$ which corresponds to it, i.e., the $i$-fold application of successor to the element $0$. This sort of thing never happens in "ordinary" mathematics, but it happens in logic precisely because in logic we do study meta-level properties of set theory.
Lastly, let me mention that there are conservative extensions of ZFC in which classes are part of the theory proper, for example the Von Neumann-Bernays-Gödel set theory. Such theories make handling of classes quite a bit easier, so perhaps we should switch over to them, lest we get entangled in technicalities concerning meta-level and object-level numbers.