Ordered tuples of proper classes

Question

From time to time I encounter notation like this:

A triple $\langle \mathbf{No}, \mathrm{<}, b \rangle$ is a surreal number system if and only if ...

The confusing part is that a proper class is used as a component of an ordered tuple (that eventually supposed to have some encoding via nested sets à la Kuratowski pair). Thus, a proper class becomes an element of a set, that is formally impossible.

So, I suppose, this is some sort of abuse of notation. What would be a formally correct way to express such things (in $\sf ZFC$, for example)?

Answer 1

Let $\langle a,b \rangle=\{\{a\},\{a,b\}\}$ denote a Kuratowski pair. Suppose we have $n$ classes (some of them can be proper classes):$$C_i=\{x\mid\phi_i(x)\},\ 1\le i\le n.$$ Our goal is to find a class that could represent an ordered tuple of these. We can use the following class for this purpose: $$\langle\!\langle C_1,\,...,\,C_n\rangle\!\rangle=\{\langle0,n\rangle\}\cup\bigcup_{1\le i\le n}\{\langle i,x\rangle\mid x\in C_i\}.$$ It is a proper class iff any of its components $C_i$ are proper classes, but all its elements are sets, tagged so that we are able to unambiguously reconstruct the length of the tuple and each of its components: $$C_i=\{x\mid \langle j,x \rangle\in\langle\!\langle C_1,\,...,\,C_n\rangle\!\rangle\land i=j\}$$

Answer 2

There are several ways to "correct" this "abuse". It should be noted that the whole point of abuse of notation/language/etc. is that you can fix these within the context of a proof, but not necessarily in one way. So different ways would yield different proofs.

If $A,B$ and $C$ are proper classes defined by $\varphi_A,\varphi_B,\varphi_C$ respectively, then we can consider the class defined by the disjunction of these three classes, and then we pick up the elements from whichever predicate we prefer by limiting our search to the appropriate class/definition.

Another way to think about it would be to consider any formal statement about $\langle\mathbf{No},<,b\rangle$ as a statement in which we will always require the first coordinates to be from $\mathbf{No}$, the second coordinates to be from a formula defining a linear order, and the third formula defining $b$>

There are probably a few more ways to think about this formally. One only has to stop and think about the essenece of proper classes in $\sf ZFC$, and then the context in which such abuse of notation is used becomes clear.