As I was reading through some wiki articles, as well as my texts, I found the following two definitions for the ordered pairs and $n$-tuples:
$(a,b):=\{\{a\},\{a,b\}\};\quad (a_1,a_2,\ldots,a_n):=(X,Y,F)$,
where in the second equation an $n$-tuple is represented as a function from $X$ (whose cardinality is $n$) to $Y$, via the rule $F$.
Okay... but here is my question: it seems to me that in order to define what a function is, which is itself a triple, I must devise a way to define such triple. Fine, one can extend the definition of an ordered pair to define triples, or any $n$-tuple for that matter, so how do we reconcile this fact? I mean, how does one define in the first place a triple, so that one can meaningly talk about functions, and use that very thing to define... what is exactly the same thing? I am lost.
So mainly my question is as follows: if one uses a triple to define $n$-tuples, how does one define that triple? If there exists another way to define that triple, why then one would bother using functional definition?
Thanks bunch in advance!
The functional definition does rely on the pairing definition, but they are not equivalent in all cases. The functional definition is needed to talk about infinite tuples. For example, a function $f: \mathbb{N} \to A$ can be thought of as a $\aleph_0$-sized tuple of elements of $A$. This tuple cannot be created using the first definition alone, since any number of applications of pairing still results in a finite set.
For example, normally for sets A and B, you'd define the cartesian product $A \times B = \{(a,b) : a \in A, b \in B\}$, and this can easily scale up for any finite number of sets. But if we have an infinite indexed family of sets $F_i$ and wish to define the cartesian product $\prod_{i \in I} F_i$, you can't use pairing to create such elements and instead need to define each element as a function $f: I \to \bigcup F_i$ such that $f(i) \in F_i$.
So you use the pairing definition to define the triple
$$ (X,Y,F) = ((X,Y),F) = \{\{X, \{X,Y\}\}, \{\{X, \{X,Y\}\}, F\}\} $$
which itself may define a function that represents an infinite tuple.