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

On 8 Sep 2005 13:50:59 -0700, "Timothy Golden
http://www.BandTechnology.com"; <tttpppggg@xxxxxxxxx> wrote:

>
>David C. Ullrich wrote:
>> On 8 Sep 2005 10:22:24 -0700, "Timothy Golden
>> http://www.BandTechnology.com"; <tttpppggg@xxxxxxxxx> wrote:
>>
>> >
>> >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.
>>
>> Uh, thanks. I understand real analysis very well. The formulas
>> above do not define a bijection from the plane to the line,
>> or in the other direction.
>>
>> Something that has a range coming close to every point in a set
>> is not a mapping _onto_ that set. Saying "this is the epsilon-delta
>> method of thinking" does not change that fact.
>>
>> >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.
>>
>> A spiral is not a space-filling curve at all.
>
>Most of the space filling curves that I have seen fill a box of finite
>measure.
>This one carries out to infinity so ought to be granted similar status
>as one which accomodates a box with zero error.

Sorry, but words mean what the definitions say they mean - your
feelings about what should be accorded what status have nothing
to do with whether a given statement is true. You're free to
feel that a spiral "should be regarded as" a space-filling curve;
that has no bearing on the fact that it _is_ _not_ a space-filling
curve.

>-Tim
>
>> And in fact it's very easy to see that a bijection from
>> R to RxR _cannot_ be continuous. So those formulas above
>> can't possibly be right.
>>
>> >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
>> >
>>
>>
>> David C. Ullrich


************************

David C. Ullrich
.

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