TITLE: Why choose sets to be the primitive objects in mathematics rather than, say, tuples?
QUESTION [14 upvotes]: Sets are defined in such a way that $\{a,a\}$ is the same as $\{a\}$, and $\{a,b\}$ is the same as $\{b,a\}$. By contrast, the ordered pair $(a,a)$ is distinct from $(a)$, and $(a,b)$ is distinct from $(b,a)$.
Intuitively, it would seem useful to draw a distinction between two collections if they are ordered differently, or if one collection has a different number of copies of an element to the other. For instance, this would mean that the collection of prime factors of $6$ would be different to that of $12$. However, it is the set, rather than the tuple, that is chosen as the primitive object. Why is it useful for the foundations of mathematics that sets have very little "structure", and would their be any difficulties in choosing tuples to be the primitive object instead?

REPLY [4 votes]: Sets can be infinite, and there are different cardinalities of infinite sets; we know by Cantor's diagonal argument that there are "more" real numbers than natural numbers, so not every set can be indexed by the natural numbers. Therefore, many sets including the set of real numbers simply cannot be represented as an infinitely long tuple.
There's some talk in the comments about how you could define tuples to be indexed by an arbitrary ordinals, but in that case it seems to me that infinite tuples would not be primitive objects at all; they would be defined in terms of other things (ordinals and functions).