As you probably know, you can define $2$-tuples $(x_1,x_2)$ as $\{\{x_1\},\{x_1,x_2\}\}$; then you can define $n$-tuples $(x_1,x_2\ldots,x_{n})$ as $((x_1,x_2\ldots,x_{n-1}),x_n)$.

In alternative, you can define ordered pairs $\langle x_1,x_2\rangle$ as $\{\{x_1\},\{x_1,x_2\}\}$ (please notice the use of "ordered pairs" instead of "$2$-tuples" and the use of angular brackets instead of round ones); then you can see $n$-tuples as finite sequences, that is functions whose domain is the set of natural numbers from $1$ to $n$ and whose codomain is the set $\{x_1,\ldots,x_n\}$. So $n$-tuples are sets such as $\{\langle 1,x_1\rangle,\ldots\langle n,x_n\rangle\}$; $0$-tuples are defined to be the empty set.

The first definition is not so rigorous (see the use of dots) and works only for $n\geq 2$. The second definition is rigorous and works for every $n$, but then you end up having ordered pairs and $2$-tuples being different objects; this also implies that you have two kind of cartesian products, two kind of binary relations, two kind of functions and so on.

Is there a way to avoid such problems? Is there another better definition for $n$-tuples?

Thanks.

lengthof a finite tuple is not determined, when the x_i are themselves tuples, because you don't know how much to "unwrap" it. So this formulation is unsuitable, for example, in a context when one wants to consider the set of all finite tuples of a set of tuples. $\endgroup$ – Joel David Hamkins Jan 26 '10 at 13:09