# Defining ordered tuples of arbitrary length in NBG

How can we define in NBG or in any other axiomatization of set theory that uses classes, the concept of ordered tuples of proper classes. For example the classical Kuratowski definition does not work since if $$X$$ is a proper class, then $$\{X\}$$ is the empty set/not defined. However we can get around this by saying that an ordered pair of proper classes is defined like this $$T=(X, Y)=\left\{ \emptyset \right\} \times X \;\cup \left\{\left\{ \emptyset \right\}\right\} \times Y$$, where the cartesian product is defined in terms of ordered tuples of sets(such as Kuratowski's definition). From the definition above we can clearly recover the first/second component of the ordered pair, $$X = \text{range}(T \cap (\{\emptyset\} \times V)), Y = \text{range}(T \cap (\{\{\emptyset\}\} \times V))$$, where $$V$$ is the universal class, and $$\text{range}(A)$$ is the range of the relation $$A$$.

We can generalize this for ordered tuples with finite length by taking $$(Y_1, Y_2, ..., Y_{n+1}) = (Y_1, ..., Y_n) \cup \left\{ n \right\} \times Y_{n+1}$$ and the ordered pair of length $$0$$ is the empty set.

However, this approach fails when we talk about "ordered tuples" of infinite length. When talking about sets an infinite ordered pair of $$\kappa$$ elements can be considered as a function with the domain the cardinal $$\kappa$$ and the codomain the class of all possible "outputs". This however fails because we cannot define $$\Pi_{i \in \kappa} i \times Y_i$$, because $$(Y_i)_{i \in \kappa}$$ is not defined since it would be exactly what we are trying to define.

• They aren't really ordered pairs when they don't have two elements. The term should be "tuple". Commented Feb 22 at 11:32
• The way you've extended to $n$-tuples doesn't generalize. Functions are just a kind of relation so if you can define relations you're done. Commented Feb 22 at 12:37
• $\varnothing\times X = \varnothing$, so $"T=(X,Y) = (\varnothing\times X)\cup (\{\varnothing\}\times Y)$" yields $\{\varnothing\}\times Y$. Your "definition" makes an ordered $1$-tuple always empty. Commented Feb 22 at 20:40
• Yes, I corrected right now. Made the same typo in multiple places Commented Feb 22 at 20:45