Does there existes a 'Tuple theory'?

Question

When you search for set-theory on google, you can find many relevant results about it. Set theory defines many operations on sets like compliment, union, intersection etc.

But searching tuple-theory did not yield any relevant results. Does there exists such similar concept? What are the operations defined on tuple? What are sum and product of two tuples? What about subtraction and division of tuples? Are there any operations defined on tuple or is it just a abstract mathematical object with no operations?

I'm not talking about tuples as in programming languages, where tuples are treated identical to lists.

EDIT: As Theo mentioned in comments, A tuple (a,b) can be represented as {{a},{a,b}}. Then $(a,b) \bigcup (c,d)$ equals {{a},{a,b},{c},{c,d}}. This does not seem to correspond to a valid tuple!!

Answer 1

Set theory exists because:

• it was realized that sets of $X$s, sets of sets of $X$s etc. are frequently convenient or even necessary in proofs about $X$s;
• the most naïve assumptions about such sets lead to paradoxes;
• axiomatic set theories such as $\mathsf{ZF(C)}$ have not only long survived attempts to find paradoxes in them, but allow us to model all older mathematics.

Early axiomatic set theories came about around the time we formalized older mathematics as a whole (OK, maybe a bit later than that, but relative to the entire history of mathematics they were similarly timed, recent events). Such formalization often shows how to model specific objects so they satisfy desired/expected axioms. This is the motive, for example, in several constructions of real numbers. Which one you pick is usually unimportant, though; all that matters is you can conveniently verify what we were always convinced of. That the choice of model doesn't matter beyond that is basically the point of model theory.

In set theory, we can construct non-negative integers, then integers, then rational numbers, then real numbers, then complex numbers etc. The simplest way to do the last part is with $2$-tuples, i.e. identify $\Bbb C$ with $\Bbb R^2$ (with suitable definitions of the former's arithmetic). $2$-tuples are also important in defining functions as sets (set theory likes to make everything a set, except when it doesn't). But more generally, what is $\Bbb R^\kappa$ for $\kappa$ a cardinal (or of intermediate ambition, when it's a positive integer)?

As I mentioned in a comment, ordinal-valued $\kappa$ admit one easy construction choice:$$(x_\beta)_{\beta\in\kappa}:=\{\{x_\gamma|\gamma\le\beta\}|\beta\in\kappa\}.$$With cardinals that aren't ordinals, we might prefer e.g. to regard a tuple as a function from elements of $\kappa$ to the necessary $x$s. Again, though, these ideas are just prove-it-works implementation choices we don't really care about.

I'll finish by discussing tuple concatenation. Let $\oplus_C,\,\oplus_T$ respectively denote cardinal addition and tuple concatenation, viz. $(x_\beta)_{\beta\in\kappa}\oplus_T(y_\beta)_{\beta\in\lambda}:=(z_\beta)_{\beta\in\kappa\oplus_C\lambda}$, where $z_\beta:=x_\beta$ for $\beta\in\kappa$, and the $y_\beta$ are construed as subsequent $z$s. (In the case of ordinal-valued cardinals, for example, $z_{\kappa+\beta}:=y_\beta$ for $\beta\in\lambda$.)