Definition of an $n$-tuple agreeing with the Kuratowski's definition of an ordered pair Is there a nice and elegant definition of an $n$-tuple ($n$ is a nonnegative integer) in ZFC, which will at the same time agree with the Kuratowski's definition of an ordered pair, i.e. $\left ( a,b \right )=\left \{ \left \{ a \right \},\left \{ a,b \right \} \right \}$? I can't seem to find one.
 A: Answer corresponding with comment of Asaf.
Define the (Kuratowski) ordered pair $\left(a,b\right):=\left\{ \left\{ a\right\} ,\left\{ a,b\right\} \right\} $ and identify
$n$ as the set $n=\left\{ 0,\dots,n-1\right\} $ (the finite ordinals) and define $n$-tuple:
$$\left\langle a_{0},\dots,a_{n-1}\right\rangle :=\left\{ \left(i,a_{i}\right)\mid i\in n\right\} $$
I deliberately use other brackets here. These sets are actually functions with domain $n$. 
Now $2$-tuple $\left\langle a,b\right\rangle $
is set: $$\left\langle a,b\right\rangle=\left\{ \left(0,a\right),\left(1,b\right)\right\} =\left\{ \left\{ \left\{ 0\right\} ,\left\{ 0,a\right\} \right\} ,\left\{ \left\{ 1\right\} ,\left\{ 1,b\right\} \right\} \right\} $$
and clearly $\left\langle a,b\right\rangle \ne\left(a,b\right)$ corresponding with remark: '..$2$-tuples are not ordered pairs'.
A: Define the (Kuratowski) ordered pair $\left(a,b\right):=\left\{ \left\{ a\right\} ,\left\{ a,b\right\} \right\} $ and identify
$n$ as the set $n=\left\{ 0,\dots,n-1\right\} $ (the finite ordinals) and define $n$-tuple:
$$\left\langle a_{0},\dots,a_{n-1}\right\rangle :=\left(\left\langle a_{0},\dots,a_{n-2}\right\rangle,a_{n-1}\right) \tag{for $n>2$}$$
and $$\left\langle a_{0},a_1\right\rangle := \left(a_0,a_1\right) $$
