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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!

Andrés E. Caicedo
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user134070
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  • You might find this discussion relevant: http://math.stackexchange.com/questions/1556187/is-the-cartesian-product-of-sets-associative/1556478#1556478 – BrianO Dec 02 '15 at 22:07

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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.

Dan Simon
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