# FOM: Canonical codings

Kanovei kanovei at wmwap1.math.uni-wuppertal.de
Mon Sep 3 16:29:41 EDT 2001

> From: JoeShipman at aol.com
> Date: Mon, 3 Sep 2001 03:18:29 EDT

> To canonically enumerate the rationals we have a problem.  The usual way to
> do it is to <...>

A canonical enumeration of the rationals must be based on some
clear mathematical idea not at all related to codings.
What comes in mind is to find a nice, meaningful action of N
on Q, which would yield us the notion of
"the next rational".

The following is a nice, meaningful action of N on Q_2
(nonnegative binary rationals).
Let q\in Q_2, say, q= 3/8.
This is naturally described by the sequence 011000000.....
Now define q', that is, the "next" one, to be the one obtained
by the dyadic addition of 1000000000... , which gives us
1110000000000000, that is, 7/8,
and then q'' will be obviously
000100000000, that is, 1/16 .
Et cetera.

One can easily fit this to Q_3, Q_5,... , maybe a good algebraist
knows how to maintain the procedure for Q as a whole or, the other way
around, show that this is not possible ?