Re: bijection of R: R <--> Rx.....xR


David C. Ullrich wrote:
> On 7 Sep 2005 10:21:28 -0700, "Timothy Golden
> http://www.BandTechnology.com"; <tttpppggg@xxxxxxxxx> wrote:
>
> >Does anyone reject this method on philosophical grounds?
> >The digits are merely a representation of a real number,
> >not the real number itself. A value (a) and (b) in the reals
> >would seem more valid, and a function defined mathematically:
> > c = f ( a, b ).
>
> First, it seem like _you_ are wrongly rejecting something
> on philosophical grounds: Although it turns out it doesn't
> quite solve the problem, if it did solve the problem there
> would be nothing wrong with defining a function f(a,b) in
> terms of the decimal digits.
>
> >This thing you guys are doing is sort of a three tape Turing solution.
> >Yes it works but where is the purity?
> >How about a swirl where
> > t = c
> > r = c d
> >where t is theta and r is radius.
> >now a = r cos t
> >and b = r sin t
> >Within a delta related to d there will be a range of c that matches for
> >any a and b.
> >If more accuracy is needed then drop d.
>
> First, I don't follow your definition at all. But more important,
> it seems clear that you're _not_ defining a function! You say do
> this, then you get a _range_ of c, if more accuracy is required
> do something else...
That is the epsilon-delta method of thinking isn't it? This is at the
foundation of real analysis. When you prove that for any range delta
you can choose an epsilon that suffices you have proven the general
situation. However small you want the error that sets d in the swirl
construction above. Choosing d = 1 gets a swirl emanating from the
origin passing through 1,2,3,... on the complex plane. Based on a
single unsigned continuous value two real values can be generated(with
error). It is the simplest space filling curve. Whether the approach
can be generalized to three real values(3D) I'm not sure.
>
> To define a function f(a,b) you need to say exactly what f(a,b)
> _is_ (which the definition in terms of digits does!), not what
> it might be, or what it is approximately.
>
> >Does this approach work for 3D?
> >I don't see it.
> >
> >-Tim
>
>
> ************************
>
> David C. Ullrich

.

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