What does "equipotent sets are indistinguishable" mean? Ed Royden's Real Analysis reads "two sets that are equipotent from set-theoretical point of view are undistinguishable". I believe this assumption allows to equate, for instance, cartesian products $N^3$ and $N^2 \times N$ when cartesian product is defined as a set of all mappings. Then what stops us from equating $\{1, 2, 3\}$ and $\{2, 3, 4\}$ as clearly there is one-to-one correspondence between them. From what point of view are they different?
 A: I have never studied foundations, so please take what I am about to say with a pinch of salt. In my opinion and in my experience of studying mathematics, the majority of the time we consider equipotent sets indistinguishable if the two sets are related in some natural way, that it makes sense to identify them.
For instance, $\Bbb R^3$ does not literally have the same elements as $\Bbb R\times\Bbb R^2$, but most authors would set them equal without further comment. Yes they are equipotent, but importantly they represent the same essential idea. As a three-point set, I consider $\{1,2,3\}$ and $\{2,3,4\}$ the same, as I would $\{\text{apples},\text{pears},\text{stairs}\}$. What do the elements even represent? But, if we were doing a computation, or if working on a proof where the number $1$ plays a vitally different role to the number $4$, say, then we certainly would not consider them indistinguishable. They may be isomorphic sets, but we do not always find it convenient or correct to treat isomorphism as equality. The same goes for isomorphic groups, fields, rings, etc. - there is a time and a place to consider them equalities.
Under certain conditions, the many Riesz representation theorems establish isomorphism between linear functionals on a (measure) space and Radon measures, or even just elements of the (inner product) space itself. There is an equipotency, but no one would call these different sets indistinguishable. The Riesz theorems are nontrivial and of great importance: it is usually more appropriate to consider measures and functionals as different, but to use the Riesz dualities to convert problems in one into problems in the other.
