# Frege: Defining a universe of ordered pairs?

Does anyone know how to define the set (or class) of all ordered tuples?

In Frege's Begriffsschrift a tuple $$(x, ..., z)$$ is defined as a primitive object which is provided as an argument to a function. Therefore I'm trying to define a domain and co-domain for these functions in the form $$R : U \rightarrow \{T,F,Unk\}$$ where $$U$$ is a universe of tuples.

I was considering the following as an idea, but it requires set theory and the ordered pair as a set. It also requires Kleene $$L^+$$ and possibly a $$U$$ of natural numbers. $$S_0 = \{()\}, S_1 = U$$ $$S_{n+1} = S_n \cup {S_n}^+$$ $$S =\cup_{i=0}^\infty S_i$$

$$i$$ represents the depth of the object.

$$U = \mathbb N$$