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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.

$U = \mathbb N$

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