Enumeration of all positive fractions recently has gained renewed interest (see the list below).
By translation invariance we can be sure that in all intervals (n, n+1] of the real axis, there are the same number of fractions: #(n, n+1] = #(m, m+1] for all natural numbers n and m.
That is a matter of symmetry, independent of the method used for counting them.
Cantor's famous sequence
1/1, 1/2, 2/1, 1/3, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 5/1, 1/6, ...
however has half of its hits in the first unit interval and less than 1/n in the interval (n, n+1]. This raises two important questions: (1) What are the precise values of the function f(n) = #(n, n+1]/#(0, 1] in the limit? (2) how can the deviation from f(n) = 1 be interpreted?
None of these question could be answered in MathOverflow or Math.StackExchange.
My question: Can it be that this topic has never been treated in the literature or in lessons on set theory, for instance in the first years of set theory? I don't believe this but could not yet find any evidence.
List of sites
What fraction of fractions does Cantor's famous sequence enumerate? https://mathoverflow.net/questions/362791/what-fraction-of-fractions-does-cantors-famous-sequence-enumerate
Relative abundance of rationals in Cantor's bijection? https://math.stackexchange.com/questions/3708845/relative-abundance-of-rationals-in-cantors-bijection
Translation invariance https://groups.google.com/forum/#!topic/sci.logic/OIrleZcHXW0%5B1-25%5D
Enumerating the rationals https://groups.google.com/forum/#!topic/sci.math/D2UYGbt9Qh8
Translation invariance https://groups.google.com/forum/#!topic/scivszfc/tt_qy_ymAgo
Enumerations of the Rationals - two methods https://groups.google.com/forum/#!topic/sci.math/uudISZF3An4