How to define tuples?

Question

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.

Answer 1

I think the truth is that nobody cares. I mean, you care about such matters a little bit while learning how set theory can be used as a foundation for mathematics, but it soon ceases to be of any importance. In practice, the one important thing about n-tuples is the relation between the n-tuple and its components, i.e., the fact that two n-tuples are the same if and only if they have the same components in the same order.

If you don't learn to stop worrying about such minutiae, you will have plenty more troubles as you learn about number systems. What is the number 3, really? It could be the ordinal {0,1,2} (i.e., {∅,{∅},{∅,{∅}}}), or it could be the integer 3 represented as an equivalence class {(m,n):m=n+3} of ordered pairs of ordinals, or it could be the rational number 3 represented as an equivalence class {(p,q):p=3q, q≠0} of ordered pairs of integers, or it could be the real number 3 represented by whatever your method of defining the real numbers happen to be, or it could even be the complex number represented as a pair of real numbers (3,0) … I hope you get my drift. Every time you expand the number system, and often when you generalize some notion or other, the new contains an isomorphic copy of the old and nobody cares to distinguish between copies.

This practice of identification has its dangers, of course, so it's good that you worry about such things a bit while learning, but expect such matters to recede into the background in order to make room for more important things.

(For what it's worth, I think the method in your second paragraph is good, but having two kinds of ordered pairs should soon stop bothering you.)

Answer 2

Some people do have to care about such details, at least in unusual contexts, and I do think it’s generally worth being aware of your foundations. The details of the definition of ordered pairs is crucial in Quine’s New Foundations (e.g., https://en.wikipedia.org/wiki/New_Foundations#Ordered_pairs), and taking it as primitive can have actual set-theoretic consequences in NF. In Church’s unpublished supplement to his “Set Theory with a Universal Set,” he uses a deliberately ugly [my interpretation] definition of m-tuple to avoid collisions. In my follow-on work, I use the usual Kuratowski definition of ordered pairs, since their internal structure allowed me to model the singleton function as a set, since it’s a 2-equivalence class, for a generalization of Church’s definition of j-equivalence relations.

Answer 3

My point of view is that there is no inherent problem in using either construction. If you adopt a modicum of categorical language, then you can define the set of ordered pairs in either way, then define 'function as subset with extra properties', define composition. Until that point you have no way of comparing sets, so cannot say within the language that the two Cartesian products are different (Can one say different before one can say 'the same'?) The categorical point is then that 'product' is defined by a universal property and so is determined up to isomorphism (bijection) only, hence having two different models with the same property is no big deal.

You may not want to introduce categorical language, but realising there is no problem and that set theoretic ideas cannot tell the difference between two 'different' but bijective sets seems to be a step towards a solution to your conundrum.

Answer 4

If you really want to get into you can also check Nicolas Bourbaki "Théorie des ensembles". Category theory can also give an answer on this question, but i think it's too much for such question. But if you use set-theory as an instrument, you can just lay on the indistinguishability of isomorphic entities up to a choosen algebraic system (as others said).