Re: udev and devfs - The final word

[Hugo Mills <hugo-lkml@carfax.org.uk>]

[-- Attachment #1: Type: text/plain, Size: 1745 bytes --]

On Mon, Jan 05, 2004 at 09:52:45AM -0800, Davide Libenzi wrote:
> On Mon, 5 Jan 2004, Vojtech Pavlik wrote:
> 
> > On Mon, Jan 05, 2004 at 08:13:26AM -0800, Linus Torvalds wrote:
> > 
> > > But the thing is, some things you simply _cannot_ number. For example, a
> > > two-dimensional space is innumerable - you need more than one integer
> > > number to look things up.  So is the set of real numbers (but not the set 
> > > of fractions), etc etc.
> > 
> > Two dimensional discrete space (*) is enumerable. Just start at [0,0]
> > and assign numbers going around the center in a growing spiral (**).
> > That way you assign a number to every point in that space. This is very
> > similar to the trick used to demonstrate fractions are enumerable.
> 
> Vojtech, a spiral (in the math sense) won't work because whatever 
> continuos function you choose for the radius, you are going to skip 
> integers when the radius grows (and duplicate them when it's small). Also, 
> IIRC, fractions are enumerable because they're a mapping from two 
> enumerable spaces (integers): F = F(I1, I2) = I1 / I2.

   I think he meant something like this:

( 0,  0)
( 1,  0)
( 0,  1)
(-1,  0)
( 0, -1)
( 2,  0)
( 1,  1)
( 0,  2)
(-1,  1)
(-2,  0)
(-1, -1)
etc.

   Rationals are countable since they're the product of the integers
(numerator) and the natural numbers without zero (denominator). You
can count them in a similar way to the above "spiral", making sure
that you don't count 1/2 and 2/4 as two different numbers. :)

   Hugo.

-- 
=== Hugo Mills: hugo@... carfax.org.uk | darksatanic.net | lug.org.uk ===
  PGP key: 1C335860 from wwwkeys.eu.pgp.net or http://www.carfax.org.uk
      --- Try everything once,  except incest and folk-dancing. ---      

[-- Attachment #2: Digital signature --]
[-- Type: application/pgp-signature, Size: 189 bytes --]