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Conway's box function is the inverse of Minkowski's question mark function. It maps the dyadic rationals on the unit interval to the rationals using the Stern-Brocot tree (Farey sequence). When the rational numbers is used as the domain, it appears that the range is the quadratic field. Can we generalize this relationship?

On the unit interval, call the dyadic rationals Box Set 0, the rationals Box Set 1, the quadratic field Box Set 2, and in general, when Box Set K is the domain, call the range mapped by Conway's box function Box Set K+1.

Question 1: What is the rule for determining Box Set N for a given natural number N?

Since each Box Set is countably infinite (Aleph Null), and the real numbers on the unit interval are not countably infinite (at least Aleph One), there must be a set of the real numbers which will never be contained in any Box Set N as N goes to infinity. We may call that set the "unboxables".

Question 2: What is the "unboxable" set? (Or conversely, what is the "boxable" set: Box Set N as N goes to infinity?)

(Note also that applying the Question Mark function will also produce an infinite set of progressively "sparser" yet still countably infinite sets of rational numbers as Box Sets -1 through -N as N goes to infinity, raising related questions.)

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  • $\begingroup$ That the real numbers have cardinality $\aleph_1$ is not known, but rather is an assertion known as Cantor's Continuum Hypothesis. This hypothesis has been proved to be independent of the usual axioms (Zermelo-Frankel with Choice). $\endgroup$ Commented Jun 23, 2015 at 0:02
  • $\begingroup$ The real numbers have cardinality at least $\aleph_1$. There are proper subsets of the reals of cardinality $\aleph_1$, but that adds no further information. Given any countably infinite set of reals, there are reals that aren't in that set, but that again adds no further information. I'm not sure whether that settles the points you raise in your comments. In any event, there are dozens of expository accounts of orders of infinity on the web and in the libraries, and I encourage you to avail yourself of them if you wish to learn more about the matter. $\endgroup$ Commented Jun 25, 2015 at 0:51
  • $\begingroup$ It's coming back to me from 1967 or so... Cantor's hypothesis is that a cardinality of 1 is sufficient for the real numbers. In that case there are uncountably many Unboxable numbers. I would be interested in any work putting bounds on either side of Box Set 3. $\endgroup$ Commented Jun 25, 2015 at 4:50
  • $\begingroup$ What kind of bounds do you have in mind? In terms of cardinality, countably infinite is countably infinite. All countably infinite sets have the same cardinality. $\endgroup$ Commented Jun 25, 2015 at 5:39
  • $\begingroup$ I've corrected "Aleph 1" to "at least Aleph 1" which does not change the argument that Unboxables exist. When I say "bounds" in this case, I am not referencing cardinality, as the Box Sets form a nested hierarchy, all countable, and each set a proper subset of the next. I'm wondering about the strongest provable assertions as to what type of numbers must be included in Box Set 3 (besides the quadratic field) and as to what cannot be included. I don't know if continued fractions can still offer clues, or whether another approach is needed. I lack the skill set to make much progress. $\endgroup$ Commented Jun 25, 2015 at 12:46

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