List of explicit enumerations of rational numbers

Question

A well-known mathematical fact is that the rational numbers are countable, i.e. there is a bijective function

$$f:\mathbb{N}\rightarrow \mathbb{Q}$$

I am interesting in making a list of all explicit such bijections since each one that I know have a different philosophy behind it.

This is one of the more counterintuitive facts about infinite, at least when one enters inside set theory. When I explain this in the first time to anyone, he/she is surprised. Thus I think it will be useful for showing the fact in a more clear manner or as possible exercises to show alternatives ways with respect to the standard one.

possible duplicate of Produce an explicit bijection between rationals and naturals? — lhf, Jun 19 '13 at 17:20
This is not a duplicate, as he is asking a list of all such bijections. — Calvin Lin, Jun 19 '13 at 17:45
I doubt that you will get any sensible answer. Note that it is sufficient to find 'explicit' bijections from $g:\mathbb{N} \rightarrow \mathbb{N}$, and just apply $f g $ for a fixed $f$. This might be more intuitive to you, though it hides what happens in $fg$. — Calvin Lin, Jun 19 '13 at 17:47
Such a "list" would have uncountably many entries IF one includes only one bijection in each item on the list. But maybe there are some interesting ways of classifying the various sorts of bijections between these two sets. — Michael Hardy, Jun 19 '13 at 18:16
I hope to see if there are different approaches distinct from the usual ones, in this way obtaining essentially different ideas for such a bijection. — Josué Tonelli-Cueto, Jun 19 '13 at 18:37

Josué Tonelli-Cueto · Answer 1 · 2013-06-19T19:00:55.440

For an example I like a lot the following one based on the fundamental theorem of arithmetic, which is different from the usual approach using the plane.

Consider the map $T:\mathbb{N}\mapsto \mathbb{Z}$ given by $$T(n)=(-1)^n\left\lfloor \frac{n+1}{2}\right\rfloor$$ which is a bijection from $\mathbb{N}=\{0,1,\ldots\}$ onto $\mathbb{Z}$. Then the fundamental theorem of arithmetic gives the following nice bijection $R:\mathbb{N}_1\rightarrow \mathbb{Q}_{>0}$ given by $$R(n)=\prod_{p\text{ prime}}p^{T(\nu_p(n))}\text{,}$$ where $\nu_p$ is the $p$-adic valuation.

Joining this two we can have the desired bijection $f:\mathbb{N}\rightarrow \mathbb{Q}$ given by $$f(n)=\begin{cases} 0 &\text{if }n=0\\ (-1)^nR(|T(n)|)&\text{if }n\neq 0\\ \end{cases}$$ which can be written as an explicit formula, but gives no new information.

This and other kind of constructions I am looking for. – Josué Tonelli-Cueto Jun 19 '13 at 18:55 — Josué Tonelli-Cueto, Jun 19 '13 at 18:55

List of explicit enumerations of rational numbers

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