 Summary
 I posted earlier regarding a corollary (regarding the rational numbers) of the theorem in question here. Upon reviewing the chapter and putting pen to paper, however, I have come across a deeper question concerning the proof presented of the theorem that a countable union of countable sets is itself countable. A photo of the theorem statement and proof in question is, again, attached below.
My question concerns the portion of the proof stating, “...we set up a correspondence between the elements of U(A_n), for n in N, and a subset of S by making the element a correspond to (m, n) if A_m is the first set in which a appears, and a is the nth element of A_m.”
In particular, I am wondering whether a more precise, or rigorous/symbolic formulation of the quoted portion of the proof above can be made? In other words, how can a statement of the correspondence be formally mathematically expressed (i.e., explicitly in functional form)?
It seems (to me, at least) this would be a function mapping from U(A_n) into a subset of N x N, since, in turn, the correspondence between (m,n) and N is priorly given as a formula in the proof: (1/2)(m+n2)(m+n1) + n.
*Update: I’ve done a bit of research into the matter since initially posting, and it seems the solution may involve the axiom of choice (which I should, regardless, brush up on) as well as the particular correspondences denumerating each A_n. I’ll post any tentative solution I reach, should I arrive at one before any helpful or suggestive responses!
In particular, I am wondering whether a more precise, or rigorous/symbolic formulation of the quoted portion of the proof above can be made? In other words, how can a statement of the correspondence be formally mathematically expressed (i.e., explicitly in functional form)?
It seems (to me, at least) this would be a function mapping from U(A_n) into a subset of N x N, since, in turn, the correspondence between (m,n) and N is priorly given as a formula in the proof: (1/2)(m+n2)(m+n1) + n.
*Update: I’ve done a bit of research into the matter since initially posting, and it seems the solution may involve the axiom of choice (which I should, regardless, brush up on) as well as the particular correspondences denumerating each A_n. I’ll post any tentative solution I reach, should I arrive at one before any helpful or suggestive responses!
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