Why is the recursive definition of $n$-tuples more convenient in Enderton's logic book? In Enderton's A Mathematical Introduction to Logic, he defines $n$-tuples recursively using ordered pairs, i.e. $\langle x_1,\dots,x_{n+1}\rangle=\langle\langle x_1,\dots,x_n\rangle, x_{n+1}\rangle$. But he also notes,

Finite sequences are often defined to be certain finite functions, but
the above definition is slightly more convenient for us.

I believe he's referring to the other definition (which I prefer) of $n$-tuples as functions.
Here is a related question that provides some more background, but I'm not confused about the mathematics of Enderton's definition. I'm curious about why he feels it's slightly more convenient when it feels the opposite to me.
In what way might Enderton's choice be slightly more convenient, at least in the context of his logic book?
If you have access to the book, the relevant pages are page 4 and remark 5 on page 15.
 A: The author suggests the definition of n-tuples as nested ordered pairs,
$$\langle x_1,\dots,x_{n+1}\rangle=\langle\langle x_1,\dots,x_n\rangle, x_{n+1}\rangle ,$$
has several cognitional and operational features, which might not be so significant for others. We realise from his scattered remarks that he pays heed of such features. For example, he writes on p. 209:

Recall that any function $f:\mathbb{N}^k\rightarrow\mathbb{N}$ is also
a $(k + 1)$-ary relation on $\mathbb{N}$:
$$\langle a_1,\ldots, a_k, b\rangle\in f\Longleftrightarrow f(a_1,\ldots,
> a_k) = b.$$
At one time it was popular to distinguish between the function and the
relation (which was called the graph of the function). Current
set-theoretic usage takes a function to be the same thing as its
graph. But we still have the two ways of looking at the function.

For one thing, this definition is a straightforward case of effective (computable) enumerability; see the section 'Recursive Enumerability' (p. 238 ff.)
Another is that $\langle x_1,\dots,x_{n+1}\rangle$ is an expression of a relation that describes a predicate in a domain of discourse. By this type of definition, we can partially interpret and obtain meaningful expression a predicate. For example, suppose
$$\langle x_1, x_2, x_3\rangle=\langle\langle x_1, x_2\rangle, x_3\rangle$$
We can employ this expression meaningfully as describing a dyadic predicate $P(a, x_3)$ that we can operate on obtained from a triadic predicate $P(x_1, x_2, x_3)$. Hence, we can sentences $\forall x_3 P(a, x_3)$ or $\exists x_3 P(a, x_3)$, which may be useful, particularly in studies with philosophical streak.
An interesting point to pursue (understandably not mentioned in the book) is the intrinsic connection the definition bears to Currying (I'd like to say "Schönfinkelisation" to be historically correct):
$$f(x_1,\dots,x_{n})\rightarrow (f'(x_1,\dots,x_{n-1}))(x_{n})\rightarrow (f''(x_1,\dots,x_{n-2})(x_{n-1}))(x_{n})\rightarrow\ldots\rightarrow((\ldots ((f^{n-1}(x_1))(x_{2})\ldots(x_{n-1}))(x_{n})$$
which is an important technique also in such other fields as computer programming and linguistics.
A: Actually, you don't need the definition of function by using the above recursive definition. For this, you start with induction base for $n=2$:
$(x,y) = \{\{x\},\{x,y\}\}$ (ordered pair) is a set and so all $n$-tuples are sets.
