FOM: Canonical codings

Kanovei kanovei at
Mon Sep 3 16:29:41 EDT 2001

> From: JoeShipman at
> 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 ? 

Vladimir Kanovei

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