If there exists a bijection $\varphi$ from $A$ to $B$, to what extent are the two sets interchangeable? I am seeking understanding of part of Proposition 6.7 from page 48 of Analysis I by Amann and Escher.
Quote:



Comments and questions:
The part that I don't follow is how we can assume "without loss of generality" that $X$ is the natural numbers and $A$ is an infinite subset thereof.
I am somewhat familiar with the idea that a (group) isomorphism, as special type of bijective function between groups, "preserves structure" in the sense that if you are unconcerned with the specific details of each group (such as the names of the elements), then the groups can be considered interchangeable. That is what the text reminds me of.
Is it that, whatever the names of the elements of $X$ and $A$, since they can be put into bijection with $\mathbb N$ and an infinite subset thereof respectively, we could assign an arbitrary order to the elements of $X$, and then we might as well be working with $\mathbb N$?
I appreciate any help.
 A: As far as they go, the Proposition you quote and your interpretation are correct. But the interesting fact(s) about the other set $X$ may include, for instance, some algebraic structure, which the “naked” set $\Bbb N$ may not possess.
For instance, if $X=\Bbb Q$, the field of rational numbers, then it has its own addition and a multiplication besides, which have nothing whatever to do with the addition on $\Bbb N$, no matter how the bijection $\varphi$ is cooked up. In this case, the set $X$ is most certainly not “interchangeable” with $\Bbb N$.
The set-theoretic equivalence here is the crudest possible way of saying two sets are equivalent, and may be useless for many (most, in fact) mathematical purposes.
A: It is a good intuition-guiding principle that sets of the same cardinality are interchangeable when it comes to cardinality-related properties of sets (just as you have noted that isomorphic groups are interchangeable when it comes to group-theoretic properties).
In addition to that intuition, there are also rigorous proofs that demonstrate that interchangeability in various specific ways.
In this case, there is an unspoken step: given $A\subseteq X$ and the bijection $\varphi\colon X\to\Bbb N$, define $B=\varphi(A)$ to be a particular subset of $\Bbb N$. Then the cardinalities of $A$ and $B$ are the same (since $\varphi$ remains a bijection when restricted to the domain $A$ and the codomain $B$), and so it suffices to prove that $B$ is countable. In particular, it suffices to prove that any infinite subset of $\Bbb N$ is countable, which is how the given proof is structured.
(I'll also note that another intuition about countable sets is that they can be assigned an order; however, such an ordering doesn't play any part in the unspoken step above.)
